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Standard metrics for gear condition monitoring

`gearMetrics = gearConditionMetrics(X)`

`gearMetrics = gearConditionMetrics(T)`

`gearMetrics = gearConditionMetrics(___,Name,Value)`

`gearMetrics = gearConditionMetrics(T,sigVar,diffVar,regVar,resVar)`

`gearMetrics = gearConditionMetrics(___,'SortBy',sortByValue)`

`[gearMetrics,info] = gearConditionMetrics(___)`

returns the gear condition monitoring metrics `gearMetrics`

= gearConditionMetrics(`X`

)`gearMetrics`

using the
vibration data in cell array `X`

.
`gearConditionMetrics`

assumes that each cell element in
`X`

contains columns of time-synchronous averaged (TSA), difference,
regular, and residual signals, in their respective order. If the signals are not in the
same order, then use `Name,Value`

pair arguments.

computes the gear condition monitoring metrics `gearMetrics`

= gearConditionMetrics(`T`

)`gearMetrics`

from
vibration dataset `T`

. `gearConditionMetrics`

assumes
that `T`

contains columns of TSA, difference, regular, and residual
signals, in their respective order. If the signals are not in the same order, then use
`Name,Value`

pair arguments.

allows you to specify additional parameters using one or more name-value pair
arguments.`gearMetrics`

= gearConditionMetrics(___,`Name,Value`

)

computes the gear condition monitoring metrics `gearMetrics`

= gearConditionMetrics(`T`

,`sigVar`

,`diffVar`

,`regVar`

,`resVar`

)`gearMetrics`

from
vibration dataset `T`

. Use `[]`

or
`''`

to skip a signal in the computation. For instance, if the data set
`T`

contains only the TSA and regular signal, use the syntax in the
following
way.

gearMetrics = gearConditionMetrics(T,sigVar,[],regVar,[])

allows you to specify the chronological order of the signal histories using
`gearMetrics`

= gearConditionMetrics(___,'SortBy',`sortByValue`

)`sortByValue`

. `NA4`

depends on the chronological
order of the vibration data since `gearConditionMetrics`

uses the
previous datasets up to the current index to compute the metric.

`[`

also returns the structure `gearMetrics`

,`info`

] = gearConditionMetrics(___)`info`

containing information about the
table or `fileEnsembleDatastore`

object variables assigned to various
signals.

**Root Mean Square (RMS)**

The root mean square (RMS) of the TSA signal is computed using the `rms`

command. For
a TSA signal *x*, `RMS`

is computed as,

$$\text{RMS}\left(x\right)\text{=}\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{x}_{i}^{2}}}.$$

Here, *N* is the number of data samples.

`RMS`

is usually a good indicator of the overall condition of gearboxes,
but not a good indicator of incipient tooth failure. It is also useful to detect unbalanced
rotating elements. `RMS`

of a standard normal distribution is 1.

For more information, see `rms`

.

**Kurtosis**

Kurtosis is a measure of how outlier-prone a distribution is. The kurtosis of a standard normal distribution is 3. Distributions that are more outlier-prone have kurtosis values greater than 3; distributions that are less outlier-prone have kurtosis values less than 3.

`gearConditionMetrics`

computes the kurtosis value of the TSA signal
using the `kurtosis`

command. The kurtosis of a sequence is
defined as,

$$\text{Kurtosis}\left(x\right)\text{=}\frac{\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}-\overline{x}\right)}^{4}}}{{\left[\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{\left({x}_{i}-\overline{x}\right)}^{2}}\right]}^{\text{}2}}.$$

Here, $$\overline{x}$$ is the mean of the TSA signal *x*.

For more information, see `kurtosis`

.

**Crest Factor (CF)**

`Crest Factor`

is the ratio of the positive peak value of the input
signal *x* to the `RMS`

value. `gearConditionMetrics`

computes the crest factor of the TSA signal using the `peak2rms`

command.

The crest factor of a sequence is defined as,

$$\text{CF}\left(x\right)\text{=}\frac{\text{P}\left(x\right)}{\text{RMS}\left(x\right)}.$$

Here, *P(x)* is the peak value of the TSA signal.

The crest factor indicates the relative size of peaks to the effective value of the signal. It is a good indicator of gear damage in its early stages, where vibration signals exhibit impulsive traits.

**FM4**

The `FM4`

indicator is used to detect faults isolated to only a limited
number of teeth in a gear mesh. `FM4`

is defined as the normalized kurtosis
of the difference signal [4]. `FM4`

of a standard normal distribution is 3.

`FM4`

is computed as,

$$\text{FM4}\left(d\right)\text{=}\frac{\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{\left({d}_{i}-\overline{d}\right)}^{4}}}{{\left[\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{\left({d}_{i}-\overline{d}\right)}^{2}}\right]}^{\text{}2}}$$

where, $$\overline{d}$$ is the mean of the difference signal *d*.

**M6A**

The `M6A`

indicator is used to detect surface damage on machinery
components. `M6A`

employs the same theory as the `FM4`

metric, but uses the sixth moment of the difference signal normalized by the cube of the
variance. `M6A`

of a standard normal distribution is 15. Hence,
`M6A`

is expected to be more sensitive to peaks in the difference signal.
`gearConditionMetrics`

uses the `moment`

command to compute `M6A`

.

`M6A`

is computed as,

$$\text{M6A}\left(d\right)\text{=}\frac{\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{\left({d}_{i}-\overline{d}\right)}^{6}}}{{\left[\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{\left({d}_{i}-\overline{d}\right)}^{2}}\right]}^{\text{}3}}$$

where, $$\overline{d}$$ is the mean of the difference signal *d*.

**M8A**

The `M8A`

indicator is an improved version of `M6A`

. It
is expected to be more sensitive to peaks in the difference signal since
`M6A`

is normalized by the fourth power of the variance.
`M8A`

of a standard normal distribution is 105. It is computed as,

$$\text{M8A}\left(d\right)\text{=}\frac{\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{\left({d}_{i}-\overline{d}\right)}^{8}}}{{\left[\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{\left({d}_{i}-\overline{d}\right)}^{2}}\right]}^{\text{}4}}.$$

**FM0**

`FM0`

is useful in detecting major anomalies in the gear meshing pattern.
It does so by comparing the maximum peak-to-peak amplitude of the TSA signal to the sum of the
amplitudes of the meshing frequencies and their harmonics.
`gearConditionMetrics`

uses a combination of `peak2peak`

and `fft`

commands to compute the
`FM0`

metric.

`FM0`

is computed as,

$$\text{FM0}\left(x\right)\text{=}\frac{PP\left(x\right)}{{\displaystyle \sum _{i=1}^{N}A\left(i\right)}}$$

where, *PP(x)* is the peak-to-peak values of the TSA signal. *A* contains the frequency-domain amplitudes at the mesh frequencies and their
harmonics, which represents the energy of the regular signal.

*A* is computed as,

$$A\text{=}\frac{fft\left(R\left(t\right)\right)}{N}$$

where, *R(t)* is the regular signal.

**Energy Ratio (ER)**

`Energy Ratio`

is defined as the ratio of the standard deviations of the
difference and regular signals [1]. It is useful as an
indicator of heavy uniform wear, where multiple teeth on the gear are damaged.

`Energy Ratio`

is computed as,

$$\text{ER}\left(x\right)\text{=}\frac{\sigma \left(d\right)}{\sigma \left(R\right)}$$

where, *d* and *R* represent the difference and regular signals, respectively.

**NA4**

`NA4`

is an improved version of the `FM4`

indicator
[3]. `NA4`

indicates the onset of damage and continues to react to the damage as it spreads and increases
in magnitude.

`NA4`

is computed as,

$$\text{NA4}\left(r,k\right)\text{=}\frac{\frac{1}{N}{\displaystyle {\sum}_{i=1}^{N}{\left({r}_{ik}-{\overline{r}}_{k}\right)}^{4}}}{{\left[\frac{1}{k}{\displaystyle {\sum}_{j=1}^{k}\frac{1}{N}{\displaystyle {\sum}_{i=1}^{N}{\left({r}_{ij}-{\overline{r}}_{j}\right)}^{2}}}\right]}^{\text{}2}}$$

where the normalization is across all vibration data sets up to the current
time *k* using the running average of variances of residual signals.

[1] Keller, Jonathan A., and P.
Grabill. "Vibration monitoring of UH-60A main transmission planetary carrier fault." *Annual Forum Proceedings-American Helicopter Society*. Vol. 59. No.
2. American Helicopter Society, Inc, 2003.

[2] Večeř, P., Marcel Kreidl, and R.
Šmíd. "Condition indicators for gearbox condition monitoring systems." *Acta Polytechnica* pages 35-43, 45.6 (2005).

[3] Zakrajsek, James J., Dennis P.
Townsend, and Harry J. Decker. "An analysis of gear fault detection methods as applied to
pitting fatigue failure data." *Technical Memorandum 105950*. No.
NASA-E-7470. NASA, 1993.

[4] Zakrajsek, James J. "An investigation of gear mesh failure prediction techniques." MS Thesis-Cleveland State University, 1989.