# gearMeshFaultBands

Construct frequency bands around the characteristic fault frequencies of meshing gears for spectral feature extraction

## Syntax

``FB = gearMeshFaultBands(FR,Ni,No)``
``FB = gearMeshFaultBands(___,Name,Value)``
``[FB,info] = gearMeshFaultBands(___)``
``gearMeshFaultBands(___)``

## Description

example

````FB = gearMeshFaultBands(FR,Ni,No)` generates characteristic fault frequency bands `FB` of gear mesh using the rotational speed of the input gear `FR` and the number of teeth on the input `Ni` and output gear `No` respectively. The values in `FB` have the same implicit units as `FR````

example

````FB = gearMeshFaultBands(___,Name,Value)` allows you to specify additional parameters using one or more name-value pair arguments.```

example

````[FB,info] = gearMeshFaultBands(___)` also returns the structure `info` containing information about the generated fault frequency bands `FB`.```

example

````gearMeshFaultBands(___)` with no output arguments plots a bar chart of the generated fault frequency bands `FB`.```

## Examples

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For this example, consider a simple gear set with an 8-toothed pinion on the input shaft meshing with a 42-toothed spur gear on the output shaft. Assume that the input shaft is spinning at 20 rpm. Construct the gear mesh frequency bands using the physical characteristics of the gear set.

```Ni = 8; No = 42; FR = 20; [FB,info] = gearMeshFaultBands(FR,Ni,No)```
```FB = 5×2 19.0000 21.0000 2.8095 4.8095 79.0000 81.0000 159.0000 161.0000 159.0000 161.0000 ```
```info = struct with fields: Centers: [20 3.8095 80 160 160] Labels: ["1Fi" "1Fo" "1Fa" "1Fm" "1Fm"] FaultGroups: [1 2 3 4 5] ```

`FB` is a 5x2 array which includes the primary frequencies `1Fi`, `1Fo`, `1Fa` and `1Fm` respectively. The structure info contains the center frequencies and labels of each frequency range in `FB`.

For this example, consider a simple gear set with an 8-toothed pinion on the input shaft meshing with a 42-toothed spur gear on the output shaft. Assume that the input shaft is driven at 20 Hz. The dataset `motorSignal.mat` contains vibration data for the gear mesh sampled at 1500 Hz.

First, construct the gear mesh frequency bands using the physical characteristics of the gear set. Construct the frequency bands with the first 3 sidebands and specify the '`Domain`' as '`order`'.

```Ni = 8; No = 42; FR = 20; FB = gearMeshFaultBands(FR,Ni,No,'Sidebands',1:3,'Domain','order')```
```FB = 15×2 0.9500 1.0500 0.1405 0.2405 3.9500 4.0500 4.9500 5.0500 5.9500 6.0500 6.9500 7.0500 8.9500 9.0500 9.9500 10.0500 10.9500 11.0500 7.3786 7.4786 ⋮ ```

`FB` is a 15x2 array which includes the primary frequencies and their sidebands.

Load the vibration data and compute PSD and frequency grid using `pspectrum`. Use a frequency resolution of 0.5.

```load('motorSignal.mat','C'); fs = 1500; [psd,freqGrid] = pspectrum(C,fs,'FrequencyResolution',0.5);```

Now, use the frequency bands and PSD data to compute the spectral metrics.

`spectralMetrics = faultBandMetrics(psd,freqGrid,FB)`
```spectralMetrics=1×46 table PeakAmplitude1 PeakFrequency1 BandPower1 PeakAmplitude2 PeakFrequency2 BandPower2 PeakAmplitude3 PeakFrequency3 BandPower3 PeakAmplitude4 PeakFrequency4 BandPower4 PeakAmplitude5 PeakFrequency5 BandPower5 PeakAmplitude6 PeakFrequency6 BandPower6 PeakAmplitude7 PeakFrequency7 BandPower7 PeakAmplitude8 PeakFrequency8 BandPower8 PeakAmplitude9 PeakFrequency9 BandPower9 PeakAmplitude10 PeakFrequency10 BandPower10 PeakAmplitude11 PeakFrequency11 BandPower11 PeakAmplitude12 PeakFrequency12 BandPower12 PeakAmplitude13 PeakFrequency13 BandPower13 PeakAmplitude14 PeakFrequency14 BandPower14 PeakAmplitude15 PeakFrequency15 BandPower15 TotalBandPower ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ ______________ ______________ __________ _______________ _______________ ___________ _______________ _______________ ___________ _______________ _______________ ___________ _______________ _______________ ___________ _______________ _______________ ___________ _______________ _______________ ___________ ______________ 0.82564 1 0.1542 0.057165 0.1875 0.011175 0.29169 4 0.055249 0.011486 5 0.0021583 0.070117 6 0.013877 1.0514 7 0.21675 0.0077621 9 0.001577 0.004752 10 0.0010282 0.012155 11 0.0025085 7.7318 7.4375 1.4057 4.2222 7.625 0.79678 0.92456 7.8125 0.1924 0.030489 8.1875 0.0060835 0.069138 8.375 0.012642 0.068649 8.5625 0.012578 2.8848 ```

`spectralMetrics` is a 1x46 table with peak amplitude, peak frequency and band power calculated for each frequency range in `FB`. The last column in `spectralMetrics` is the total band power, computed across all 15 frequencies in `FB`.

For this example, consider a simple pinion and gear set with an input shaft speed of 1800 rpm. Considering that the pinion on the input shaft has 6 teeth and the gear on the output shaft has 8 teeth, visualize the frequency bands for the gear mesh.

```FR = 1800; Ni = 6; No = 8; gearMeshFaultBands(FR,Ni,No)```

From the plot, observe the following:

• Output shaft defect frequency, `1Fo` at 1350 Hz

• Input shaft defect frequency, `1Fi` at 1800 Hz

• Assembly phase defect frequency, `1Fa` at 5400 Hz

• Gear mesh defect frequency, `1Fm` at 10800 Hz

## Input Arguments

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Rotational speed of the input gear, specified as a positive scalar. `FR` is the fundamental frequency around which `gearMeshFaultBands` generates the fault frequency bands. Specify `FR` either in Hertz or revolutions per minute.

Number of teeth on the input gear, specified as a positive integer.

Number of teeth on the output gear, specified as a positive integer.

### Name-Value Pair Arguments

Specify optional comma-separated pairs of `Name,Value` arguments. `Name` is the argument name and `Value` is the corresponding value. `Name` must appear inside quotes. You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: `...,'Harmonics',[1,3,5]`

Harmonics of the fundamental frequency to be included, specified as the comma-separated pair consisting of '`Harmonics`' and a vector of positive integers. The default value is 1. Specify '`Harmonics`' when you want to construct the frequency bands with more harmonics of the fundamental frequency.

Sidebands around the fundamental frequency and its harmonics to be included, specified as the comma-separated pair consisting of '`Sidebands`' and a vector of nonnegative integers. The default value is 0. Specify '`Sidebands`' when you want to construct the frequency bands with sidebands around the fundamental frequency and its harmonics.

Width of the frequency bands centered at the nominal fault frequencies, specified as the comma-separated pair consisting of '`Width`' and a positive scalar. The default value is 10 percent of the fundamental frequency. Avoid specifying '`Width`' with a large value so that the fault bands do not overlap.

Units of the fault band frequencies, specified as the comma-separated pair consisting of '`Domain`' and either `'frequency'` or `'order'`. Select:

• `'frequency'` if you want `FB` to be returned in the same units as `FR`.

• `'order'` if you want `FB` to be returned as number of rotations relative to `FR`.

## Output Arguments

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Fault frequency bands, returned as an `Nx2` array, where `N` is the number of fault frequencies. `FB` is returned in the same units as `FR`, in either Hertz or orders. Use the generated fault frequency bands to extract spectral metrics using `faultBandMetrics`. The generated fault bands, , are centered at the characteristic defect frequencies and their harmonics and sidebands for:

• Input shaft defect frequency, `Fi`

• Output shaft defect frequency, `Fo`

• Gear mesh defect frequency, `Fm`

• Assembly phase pass defect frequency, `Fa`

When you specify the sidebands, `gearMeshFaultBands` computes the sidebands with respect to the input and output shaft defect frequencies:

• Fault frequency bands for input gear defects with its harmonics and the first sideband at `Fi`

• Fault frequency bands for output gear defects with its harmonics and the first sideband at `Fo`

`gearMeshFaultBands` truncates negative fault frequency bands automatically and generates a warning message.

The value `W` is the width of the frequency bands, which you can specify using the '`Width`' name-value pair.

Information about the fault frequency bands in `FB`, returned as a structure with the following fields:

• `Centers` — Center fault frequencies

• `Labels` — Labels describing each frequency

• `FaultGroups` — Fault group numbers equal to the number of frequencies

## Algorithms

`gearMeshFaultBands` computes the different characteristic fault frequencies as follows:

• Input shaft defect frequency, ${F}_{i}=FR$

• Output shaft defect frequency, ${F}_{o}=\frac{Ni}{No}FR$

• Gear mesh defect frequency, ${F}_{m}=NiFR=No{F}_{o}$

• Assembly phase pass defect frequency,${F}_{a}=\frac{{F}_{m}}{\text{gcd}\left(Ni,No\right)}$

## References

[1] Lang, George Fox. “S&V geometry 101.” Sound and Vibration 33 (1999): 16-26.