Characterisation and evaluation of the harmonic content of involute gear surface deviations

Gears are fundamental precision components that are crucial to many industries, including the wind energy sector. They are traditionally analysed using profile and helix line measurements but this does not consider the majority of the contacting surface with intricate surface variations that dictate functional performance. Practical measurement and evaluation of involute gear surfaces remain underexplored. This paper uses the Fourier and wavelet transforms to evaluate measured gear surface deviation harmonic content for a single trace, a trace set, and surface analysis and describes the utility of each. The methods are first demonstrated with synthetic data and then with measured data, then the measured data is simulated in a quasi-static FE tooth contact analysis, which yields insights not solely gleaned from harmonic analysis. The uncertainty of harmonic content analysis is to be established in future work and how that effects the uncertainty of simulated functional performance should be investigated.


Introduction
Involute gears are precision components and are traditionally analysed using profile and helix line measurements but new surface measurement methods now enable intricate surface variations that dictate performance to be captured.Interpreting the results from gear flank surface measurement is challenging.However, characterisation of the gear flanks through harmonic analysis will provide further understanding of gear functional performance for noise, vibration, efficiency, micropitting and scuffing failure mode resistance.
Involute gear measurements are typically to ISO 1328-1:2018 or VDI/VDE 2612 and 2613-1 where profile, helix, and pitch deviations are evaluated [1][2][3].The deviations are compared to tolerance bands which were originally developed to characterise manufacturing machine capability rather than the functional performance of the gear, although they are strongly correlated.A tooth flank will typically only be measured with a single profile and helix which gives a sparse characterisation of the flank.The evaluated parameters of profile and helix slope errors relate to the load distribution during running but gears fundamentally contact as a pair and the combination of errors gives the true contacting surface.
Additionally, during engagement helical gears contact over an entire tooth flank surface with a contact line at an inclination related to the helix angle and it may not be sufficient to use the harmonic content of single traces as a indicator of true performance.Although extension of standard evaluations from traces to areal data has been suggested [4,5] and described in an recent overview of state of the art along with measurement technologies required [6], the harmonic evaluation of gear flank surfaces has not been investigated significantly.Use of point cloud data and the use of the harmonic transforms was investigated in [7], with the inversion method being showcased in more detail in [8].Jolivet et al compared various frequency methods to gears manufactured with grinding and powerhoning to understand the effect of the different finish conditions [9].
Additionally, it is of increasing interest to perform harmonic analysis to control noise and vibration from machine tools [10,11] and how they relate to the gear geometry [12,13].
The field of harmonic analysis assumes that any signal can be deconstructed into a superposition of many waves with different frequencies and amplitudes.Harmonic analysis of spatial geometry data is less common, particularly for non-repeating surfaces (ones that are not rotationally symmetric, and therefore processed data does not close causing a discontinuity in the signal to be processed).For example, the first mathematical tool utilised in this paper-Fourier analysis-assumes that signals are periodic in nature and therefore infinite.Methods, such as windowing, are used to improve either the frequency or amplitude calculation by limiting the effect of discontinuities at the edges but this can mask important information at the data edges.Additionally, Fourier analysis completely transforms the signal to frequency space and no position data is retained (figure 1(b)).
The short time Fourier transform (STFT) tries to combat this by dividing the signal into shorter and shorter segments then performing the fast Fourier transform (FFT) on each segment.This properly considers information about position but starts to lose frequency resolution as the segments get smaller; decreased resolution in frequency is traded for increased resolution in position (figure 1(c)).This relates to the Heisenberg uncertainty principle or the Gabor limit in signal processing [14].It is an inherent drawback of the STFT and is a key reason for the development of the discrete wavelet transform (DWT) where the there is varying resolution at different positions and frequencies (figure 1(d)).Multi-resolution analysis is common method of representing wavelet coefficients where each frequency level is reconstructed to a new signal, and all the levels can be combined to return the original signal.
The DWT is a relatively new mathematical tool and is utilised in many industries but is not well known in the gearing industry.Whereas the FFT assumes an infinite sine wave, the DWT uses a wave that is not infinite and thus characterises information about position.A wavelet does not have to be sinusoidal in nature and can be designed to recreate properties in the signal that is to be analysed.For example, it is desired to remove low frequency baseline wander from electrocardiogram signals to properly capture rhythms of interest, and the use of the Morlet or Daubechies wavelets prove very effective [15].A few example wavelets can be seen in figure 2.
Harmonic analysis is also a common tool for condition monitoring of gears, bearings and other rotating components which relate time series data to vibration data and there is much research in this field [17][18][19][20].
The objectives of the paper are to apply the Fourier and wavelet harmonic transforms to involute gear measurements and discuss their uses.Furthermore, highlighting how the different harmonic content can affect the functional performance by simulating the measured data in a FE based tooth contact analysis model.

Harmonic method introduction
This section details the methods used when applying the Fourier and wavelet transforms to gear flank surfaces.Three methods were used for each transform to help understand the benefits and limitations: • Trace analysis-single profile analysed in isolation.
• Trace set analysis-a collection of profiles analysed in isolation but considered together.
• Surface analysis-all flank data analysed concurrently.
The Fourier transforms will show amplitude-order plots which show the frequency content present in the analysed signal.When using surface Fourier analysis the angle of content can be calculated using (1) and the order delta between two points of interest.
angle of harmonic content from positive roll width axis to positive facewidth axis roll length order delta facewidth order delta total length of roll length data total length of facewidth data The wavelet transforms will show multi-resolution analysis of the reconstructed signal at different frequency bands.

Gear artefact geometry
Two geometries are analysed, synthetic geometry based off a large involute gear artefact and measured data from a harmonic content artefact (HCA), see figure 3 and table 1.
The synthetic geometry is generated and analysed using Przyklenk's holistic inversion method [8], the residuals can been seen in figure 4. The synthetic data includes content at 1th, 7th, 9th, 30th orders, and piecewise 50th order content at an angle of 59.5°, the generation is fully explained in the paper.The HCA is defined as the 15 degree right hand helix angle gear tooth on a large gear artefact that has been manufactured with methods representative of the wind energy industry, and in particular with the generator grinding method.This manufacturing method means that it is likely to include harmonic content inclined at an to the profile and helix directions which showcases the harmonic methods.

Measurement machines
A Klingelnberg P65 gear measurement machine (GMM) was used to measure the artefact and it is the UK's National Gear Metrology Lab's primary measurement machine.It resides within a temperature controlled lab to 20 ± 0.3 °C and the recent U95 profile calibration uncertainty values for this artefact are f Hα = ± 1.10 μm, F α = ± 1.50 μm, f fα = ± 1.00 μm for slope, total, and form evaluated parameters.
Each flank was measured with 40 profiles equally spaced in facewidth, with 450 points equally spaced in length of roll and were evaluated to helix and profile limits of 395-414 mm diameter, and 7.75-147.5 mm facewidth.Each profile was scanned with a 5 mm ruby sphere and filtered using the Gaussian filter described in ISO 1328-1:2013 and ISO 16 610-21 [1,21].The end effects were managed using the point symmetric reflection method described in ISO 16 610-28 and the standard cut off filter of L/30 applied, which was 0.8686 mm [22].This is the method that would be applied as standard when measuring profiles, however, if helices were measured instead then the filter is applied in the helix direction.Both methods will affect frequencies in the perpendicular direction to an unknown effect, which has been investigated.
For the GMM data there is 40 profiles which will allow for 20 orders to be evaluated in the facewidth direction.This is smaller than the filter cutoff of 30 orders, when filtering in the helix direction, essentially providing no filter at all.If 99 profiles were measured this will allow for 44 orders in the facewidth direction and essentially 14 orders that can have their amplitudes attenuated by 50% or greater by the filter.
A surface (2D) Gaussian filter would consider the effects of both profile and helix in a single filter, however, there remains an issue of how to process data content at the edges-specifically the corners.Either content at the edges gets thrown away or content is extended around the edges and corners of evaluation.The effect of filters applied to gear profile and helix measurements is discussed in depth in Reavie et al [23].

Tooth contact analysis
To showcase the effect of using measured data in a simulation a tooth contact analysis (TCA) has been performed of the data from the harmonic content artefact, using Dontyne Systems Gear Production Suite [24].
The artefact was considered a full gear and mated with a mirror gear with opposite helix angle, where the mating wheel had no involute modifications or errors applied.Three conditions were analysed: • Nominal-no modification.
• Slopes-the evaluated surface slopes from the GMM.This is a typical approach of using measurement information when available.
• GMM-the measured data from the GMM.
The TCA is a two stage model where the tooth stiffness is determined using a 3D FE model, then variations positions of tooth engagement contact are modelled statically to calculate a quasi-static FE model of the gear contact.A torque and speed of 3000 Nm and 1000 rpm were used within the analysis, with all other settings left as the default.The absolute values of the simulated results are not relevant but the trend of the relative differences between them.The parameters captured from the TCA were chosen to relate to failure modes such as micropitting, scuffing, and noise and vibration.

Results and discussion
This section details the harmonic analysis results for the synthetic data on the large involute artefact and the measured data on the HCA.

Synthetic artefact results
The data analysed is shown in figure 4. A single trace and trace set Fourier analysis is shown in figure 5.The trace analysis identifies the frequencies present in the synthetic signal at the 1th, 7th, 9th, 30th orders, but the piecewise 50th order is not clearly visible.
Figure 5(c) shows trace set analysis where content from 0th to 30th is similar for all profiles, as expected.Content around the 50th order varies across facewidth suggesting that the characteristics of the content changes as facewidth changes.This confirms reality, where the 50th order content translates in roll length as well as facewidth.As the content moves to the centre of roll length it may be better characterised than at the edges, giving the pinched appearance.Note that it is not evident that there is any parabolic crowning in the facewidth direction as the analysis is performed in roll length only.
Surface Fourier analysis is shown in figure 6.The sliced roll length orders show similar content to the single trace and trace set analysis, except the 50th order piecewise content is now a noticeable 'blip'.In fact, we can see the angled piecewise content in the surface amplitude plot and related slice plot.
For our angled content we have can take the points (0, 0) and (24, 20) which gives us L Δ = 24, b Δ = 20, L t = 300, b t = 420 and the calculated angle θ h = 59.24°from equation (1), matching with the simulated value of 59.5°.Calculating in reverse with the simulated angle gives an expected L Δ /b Δ = 24.25/20showing that the content falls between roll length orders.Single trace and trace set wavelet analysis is shown in figure 7 using a db5 wavelet at two levels.Detail level 1 contains frequencies from 50 to 100 orders with 50th order content visible at the correct position and noise content throughout.Detail level 2 contains frequencies related to with 30th order content visible for full signal length.Approximate level 2 contains the 1th, 7th and 9th order content.In the 1.5D analysis the parabolic helix crowning is also captured.
Surface wavelet analysis is shown in figure 8 using a bior3.3wavelet.The maximum levels of detail that signal can be decomposed into is related to the minimum data length.The surface data can only be decomposed into one level when using the surface analysis due to the lower data spacing in the facewidth direction compared to the roll length direction.The content analysis results look very similar to the trace set analysis except that content can be split into horizontal, vertical and diagonal directions.This demonstrates that the higher frequency content mainly changes in the horizontal (roll length) direction and the piecewise content exists in all directions.The remaining surface varies very little in the diagonal and vertical directions for this example, but it is envisaged it would prove useful when real measurement data is analysed.
To summarise the application of the Fourier and wavelet analysis methods, they are compared below for trace, trace set, and surface analysis.
• Fourier trace set Provides more information than a single trace but lacks information in the horizontal direction (figure 5(c)).
• Fourier surface Shows horizontal, vertical, and angled frequency content but is challenging to infer the size and shape of content (figure 6).
• Wavelet trace Contains frequency and position information, also splits into real valued signals using multiresolution analysis (figure 7).
• Wavelet trace set Allows for more levels of detail in the roll length direction while including some horizontal content such as crowning (figure 7).
• Wavelet surface Shows horizontal, vertical and diagonal content but is likely limited to one detail levelhowever at greater than two levels of detail there will be diminishing returns on the added value (figure 8).

Measured data
The harmonic content artefact (figure 3 table 1) was measured using the gear measurement machine described in section 2.3.A plane was fit to the surface results to isolate the harmonic content from mean slope errors.The evaluated surface slopes (the surface equivalent of the ISO 1328-1:2013 evaluated slope deviations) and an overall surface form are shown in table 2. Note that the compensation of surface slopes means that the relevant parameter to the harmonic analysis deviations is the surface form.
In figure 9 the plane leveled harmonic deviations are shown for both left and right flank.The right flank data has captured content at an angle in a single direction, whereas the left flank seems to have a combination of content in different directions.
Figures 10 and 11 shows the Fourier analysis for left and right flanks.In the trace FFT analysis content we can see spikes at the 9th and 6th order for the left and right flanks respectively, and this is continued in the trace set FFT analysis.It is difficult to interpret any angled content that may exist by looking at the trace set analysis, however, considering the surface FFT analysis we can clearly see a peak in both flanks at the 1th to 3th orders and a secondary peak 8th to 12th angled orders.Examination of the left flank results shows the two different directions of content can be seen in the surface plot, these occur both occur at the angle of the base helix angle but in different directions, and with slightly different orders.
Figures 12 and 13 shows the trace and trace set wavelet analysis of the harmonic content artefact left and right flanks.We know from the FFT results that frequencies as low as 6th order are contributing to the deviations and choosing a wavelet with a larger number of coefficients, such as db6 and higher, would include these in the approximate level.The trace wavelet analysis was performed as a 3 level decomposition with the db6 wavelet.The approximate level of the trace wavelet analysis shows different form characteristics, but the detail level show similar frequency and amplitude information, which likely relate to the machine and tool vibration characteristics.The trace set analysis was performed as a 3 level decomposition with the db8 wavelet.In both flanks at the approximate level low wavelength angled content can be seen, which mirror each other.This would have been created by the generating machining method and is at the base helix angle-confirmed by the FFT slices in figures 10 and 11.At higher detail levels the left flank primarily has content in the roll length direction whereas the right flank continues to have content at an angle in D2 and D3.This suggests that the right flank is at greater risk of affecting noise and vibration as it has combined content in the direction of contact.Due to the larger data density in the roll length direction surface wavelet decomposition is limited by the data density in the facewidth direction.Performing greater than one or two decomposition levels for the chosen wavelet would over extract content in the facewith direction, therefore a single level decomposition with the bior3.3wavelet was performed for the surface analysis in figure 14.Similar content was seen with db4 and above wavelets confirming that the limitation is data density rather than the specific content shape.The angled content is still contained within the approximate level, which limits interpretation at this stage but allows for a smoothed expression of the surface content to be transferred to a tooth contact analysis model.

Tooth contact analysis
The tooth contact analysis results are summarised in table 3, where three different micro-geometries were analysed: • Nominal-no corrections, only the nominal geometry.
• Slopes-the profile and helix surface slope data in table 2 only.• GMM-the measured data, specifically the wavelet approximate surface data in figure 8.
It can be seen that in all cases using measured data changes the simulated results for all functional parameters of interest.Considering the peak to peak transmission error (TE) the right flank measured data shows a significant twofold increase, whilst the left flank shows only a modest increase (figure 15).This confirms what was suspected from the wavelet approximate results-harmonic content direction is important.Additionally, only using the measured slope content does not capture the effect of the manufactured harmonic content.The calculated TE is quasi-static which does not consider dynamic effects, however, static TE is widely used as a metric for designing quiet gears; minimising static TE minimises dynamic TE.If the dynamic TE was calculated then the frequency content in the wavelet detail levels may be seen to have an effect in transmission error.The harmonic content is already shown to have an effect of the maximum contact stress, power losses, and contact temperature increase.

Conclusion
Fourier and wavelet analyses of synthetic tooth flank deviations were performed at different data dimensions: a single trace, a combination of individually analysed traces (trace set), and a combination of traces analysed collectively (surface).
Trace and trace set Fourier analysis provides information on the occurrences of frequencies but does not discriminate the spatial position.Extending to surface analysis gives information on the direction of frequency content, which is particularly useful for helical gears where the content is angled.Fourier analysis will be a more widely understood analysis model, however, interpreting the results and effect on functional performance is difficult, particularly for surface analysis.
Multi-resolution wavelet analysis allows for splitting of a measurement to spatial data within different frequency bands.This allows for understanding of the surface at different levels with numerous benefits: • Manufacturing process control of characteristics within a specific level.
• Measurement machine verification at different levels (form, waviness, roughness), possibly leading to multisensor gear analysis that can be combined into a single surface definition.
• Targeted data output that contains the required data fidelity for the desired parameter or feature of interest.
A harmonic content artefact has been defined which was manufactured with methods representative of the wind energy industry.The artefact was measured with a gear measurement machine and the harmonic methods applied.Understanding the effect of harmonic content in the direction of contact and related to orders of base pitch (for noise and vibration) is difficult by interpretation alone.
Simulating the measured data within a tooth contact analysis (quasi-static FE based model) provides useful information about the functional performance that may not be evident from the harmonic analysis alone.The systematic errors within measurement machines and errors that occur because of the measurement procedure  are likely to be of the same wavelength as the harmonic content of interest.Therefore, uncertainty of the harmonic content is yet to be established and should be a key focus of future work.However, perhaps considering the uncertainty of the parameters estimated in the simulation would provide a more useful description of the applicability of a measurement and would be an interesting topic of further research.

Figure 1 .
Figure 1.Position-frequency resolution plots commonly used to aid intuition of different transforms (a) Signal space-all resolution in position (b) Frequency space-all resolution in frequency (c) STFT with better position resolution but sacrificing frequency resolution (d) DWT with good frequency resolution over larger signal lengths, and good position resolution for high frequency components.

Figure 2 .
Figure 2. Example wavelets generated using the MATLAB Wavelet Toolbox [16].The first plot shows the Morlet (or Gabor) wavelet, middle plot shows the db6 scaling function, phi (f), and the right plot shows the db6 wavelet function, psi (ψ).db6 is a Daubechies wavelet with 6 vanishing moments.ψ and f are referred to as the mother and father wavelet functions respectively.

Figure 5 .
Figure 5. Synthetic Fourier analysis (a) mid profile deviations (b) Single trace FFT (c) Trace set FFT waterfall style plot where each profile has an individual FFT analysis plotted at the profile facewidth position.

Figure 6 .
Figure 6.Tooth 1 left flank deviations (a) Four quadrant surface FFT amplitudes with 0th order set to zero, surface colour is log amplitude, line colour relates to slices (refer to online content for colour plots).(b) Slice of surface FFT at roll length order = 0 (top horizontal axis), (c) Slice of surface FFT at face width order = 0 (left vertical axis), (d) Slice of angled content offset to pass through the -50 roll length order position.

Figure 7 .
Figure 7. Wavelet multi-resolution analysis for single trace and trace set from highest to lowest frequency in descending order.

Figure 8 .
Figure 8. Surface wavelet analysis of a gear, single level decomposition with with bior3.3wavelet.D-detail level, A-approximate level, superscripts d, v, and h refer to diagonal, vertical, and horizontal content respectively.

Figure 9 .
Figure 9. Measured surface deviations for left and right flanks.

Figure 10 .
Figure 10.Measured left flank Fourier analysis.(a) left top shows the mid trace, (a) left bottom shows the trace FFT results, (a) right shows the trace set FFT results.(b) top surface FFT results, bottom left facewidth slice, bottom middle roll length slice, bottom right combined slice.

Figure 11 .
Figure 11.Measured right flank Fourier analysis.(a) left top shows the mid trace, (a) left bottom shows the trace FFT results, (a) right shows the trace set FFT results.(b) top surface FFT results, bottom left facewidth slice, bottom middle roll length slice, bottom right combined slice.

Figure 12 .
Figure 12.Measured left flank wavelet multi-resolution analysis for single trace and trace set from highest to lowest frequency in descending order.

Figure 13 .
Figure 13.Measured right flank wavelet multi-resolution analysis for single trace and trace set from highest to lowest frequency in descending order.

Figure 14 .
Figure 14.Measured surface wavelet analysis of the harmonic content artefact for left and right flank.Single level decomposition with with bior3.3wavelet.D-detail level, A-approximate level, superscripts d, v, and h refer to diagonal, vertical, and horizontal content respectively.

Table 1 .
Gear parameters of synthetic data and harmonic content artefact.

Table 3 .
Tooth contact analysis results.