The traditional inspection of involute gear profile and helix deviations results in plots of deviations from a perfect involute and from a perfect helix. While this is appropriate for gears with an unmodified profile or helix, it is not ideal for gears that have intentional modifications.

Elemental inspection of gears is commonly carried out with computer numerically controlled (CNC) machines. While these range from special purpose gear measuring machines with a precision rotary table to three-axis coordinate measuring machines (CMM) with software adapted for gear measurement, what these inspection machines have in common is they produce graphical output that shows deviations of the profile from a perfect involute and deviations of the helix from a perfect helix. For gears designed without any profile or helix modifications, this is appropriate and sufficient. However, it has become very common for gears to be designed with a design profile that deviates from a perfect involute shape and a design helix that deviates from a perfect helix [1]. For these gears, a graphical output showing deviations of the profile from the design shape can be very helpful.

Background and proposal
The proper specification of design shape is very important for the proper performance of the gear. Design shape modifications may be used to improve the noise, vibration, and harshness (NVH) of a gear set, prevent tooth collisions in heavily loaded gears, compensate for changes in tooth shape due to shaft bending and thermal deformation that occur during normal operation, or for other reasons. In many gears, proper tooth shape modifications are essential for both smooth operation and long life [2]. The designer specifies what the tooth shape should be, but it is then up to the machine operator or inspector to verify that the design intent has been met. Unfortunately, for gears with modified profiles, typical gear charts require a high level of expertise to properly evaluate how the gear deviates from the design shape. What is proposed here is that since the design profile is the ideal shape, the deviations from the design profile should be displayed in addition to the deviations from a perfect involute. No additional data collection will be required; it is simply an alternate method to display the data gathered by an inspection machine. Graphical displays of deviations from the design shape are much easier to evaluate, and can be a very useful addition to the inspection processes. These deviations should, of course, be zero, so the plot of deviations from the design profile should be a straight line coinciding with the axis. These new charts are not intended to be a replacement for traditional charts, but rather an additional chart that — in combination with traditional gear charts — will make understanding of the deviations much easier.  Each method of display has its own advantages; used together they can quickly lead to a full understanding of the deviations.

Figure 1 shows a typical output of gear profile and helix (sometimes called lead) charts.  These charts show the measured traces for both left and right flanks of three teeth spaced approximately 120° apart.

Figure 2 is a schematic based on ISO 1328-1:2013 (Figure 8) showing how a single profile trace is analyzed for a gear with a design profile that has both root and tip relief. Note that the orientation of the chart is not critical as long as it is properly labeled; many charts present the traces vertically as in Figure 1 while in ISO 1328-1:2013 the traces are presented horizontally.

The same profile is shown in Figure 3, but this time the deviations from the design profile are shown rather than the deviations from a perfect involute.

Since this figure shows deviations from the design profile, the design profile will always be represented as a straight line on this figure. Therefore, no matter what the shape of the design profile, the mean profile line on this plot is a single straight line, which is the best-fit line of the deviations from the design profile over the profile evaluation range. This corresponds with the statement in ISO 1328-1 clause that “the straight-line gradient of the profile measurement is found by applying the least squares method to the deviation of the measured profile trace from the specified design profile.” With the individual deviations and the best-fit line through the deviations, it becomes very simple to find the total, slope, and form deviations.

The total profile deviation is simply the maximum deviation minus the minimum deviation from the design profile. This value is, of course, the same as that illustrated in Figure 2(a). In fact, the way Figure 2 is created is to first find the maximum and minimum deviations from the design profile, and then use these values to position facsimiles of the design profile in Figure 2(a). Most inspection machine software actually works as illustrated in Figure 3; it just does not plot it that way1. The mean profile trace in Figure 2 is found by adding the ordinates of the straight line gradient to the design profile. So again, the data needed to create Figure 2 could just as easily be used to also create Figure 3. Unfortunately, this data is normally hidden.

The form deviation is found by first calculating the deviation minus the mean profile for all the points in the evaluation range. The difference between the maximum and minimum of these values is the form deviation. Again, the value found for form deviation will be the same in both Figure 2(b) and Figure 3.

The profile slope in Figure 3 is the distance between two horizontal lines that intersect the mean profile at the profile control diameter and at the tip diameter. This can be simplified to just the difference between the values of the mean profile at the profile control diameter and the tip diameter. Of course the value found for slope deviation will be the same in both Figure 2(c) and Figure 3.

To further illustrate the concept, Figure 4 through Figure 7 show first the deviations from a perfect involute, then deviations from the design profile. In the three copies of the deviations from the design profile, the total, slope, and form deviations are shown.

In Figure 4, the design profile is a perfect involute, so there is no difference between Figure 4(a) and Figure 4(b).

Figure 5 is for a gear that has a barreled profile specified. The steepness of the rise in excess material at both the tip and root is more obvious in Figure 5(b), which shows the deviation from design, compared to Figure 5a, which shows deviations from a perfect involute. The profile slope, although it can be deduced from Figure 5(a), is very obvious in Figure 5(c) and Figure 5(d).

Figure 6 is for a gear with tip relief. The profile slope deviation again is very obvious when deviation from the design shape is shown.

Figure 7 is another example with tip relief. The display of deviation from design shape indicates that the profile is crowned with respect to the design shape.

By looking both at a figure showing deviations from a perfect involute and at a figure showing deviations from the design shape, it becomes very easy to quickly get a full understanding of the deviations.

The figures and much of the discussion has been on the profile, but the same applies to the helix. Looking at both the helix deviations from a perfect helix and at deviations from the design helix is just as beneficial as looking at both the profile deviations from a perfect profile and at deviations from the design profile.  So while the profile is discussed here, the same methodology can apply to the helix.

A suggested addition to zone-based tolerance evaluation
ISO 1328-1:2013 Annex A presents a method for zone-based tolerance evaluation. Since it often is not necessary to control a relieved area quite as closely as the central area, use of such a system can have significant manufacturing advantages.  In a zone-based tolerance evaluation, the profile (or helix) is divided into two or more zones, and each zone is calculated separately and may have different tolerance classes. For example, a profile could be divided into a tip zone, a middle zone, and a root zone. A helix could be divided into a left end zone, a center zone, and a right end zone.

The methods presented in ISO 1328-1:2013 Annex A for determining slope and form deviations in each of the zones are reasonable and do not need any modification. However, the system has three problems when it comes to total deviation:
1.   The areas between the zones are not well controlled. The areas between the zones are only considered for the plus material condition for form and total deviation.
2.   There is an immediate jump in the tolerance when entering a different zone.
3.   It is not clear how the total deviation should be applied. The text says that it is measured the same way that a gear without specified zones is measured, which would defeat the purpose of a zone-based system.

One of the notes then says that the total deviation is often only applied to the middle zone or completely omitted, so there is no control of total deviation in the end zones. If a value for total deviation is applied independently to each zone — since the tolerance bands of adjacent zones are not linked — this will result in the total allowed deviation from the design profile varying from gear to gear as shown in Figure 8(b).

These problems could be addressed with a zone-based tolerance evaluation system that specifies that the allowable total profile or total helix tolerance varies smoothly from the tolerance at the end of the central zone up to the increased tolerance at the end of the next zone. Figure 8 shows the differences between these systems.

Figure 8(a) shows a trace of deviations from a perfect involute for a profile with tip relief, with the total profile deviation shown. In Figure 8(b) the trace has been redrawn to show deviations from the design profile. If the total deviation tolerance is applied independently to each zone, then, since the tolerance bands of adjacent zones are not linked, it can be seen that if the Zone 2 tolerance is double the Zone 1 tolerance, the allowable deviation relative to Zone 1 may be anywhere from two to three times the Zone 1 tolerance and depends on the position of the trace at the beginning of Zone 2.  This may be why there is a note in ISO 1328-1:2013 A.2.3 that says, “It is common practice to restrict evaluation to the middle zone or to omit Fa completely.” Unfortunately, this means that the shape of the profile is very poorly controlled in the area where there is tip relief.

Figure 8(c) also shows a trace of deviations from a perfect involute for a profile with tip relief. In addition, it shows an example of the proposed tolerances for total profile deviation. It is critical to keep in mind that the tolerances are from the design profile, so that deviations in the positive direction just mean less tip relief, and unless they are excessive, they do not represent a plus material condition relative to a perfect involute.  It should also be realized that all the systems allow for both plus and minus deviations from the design shape; this figure just makes that fact more obvious. Note that a designer might be tempted to specify an asymmetrical tolerance increase, but it is better to achieve the same result just by changing the design profile to allow for a symmetrical tolerance.

It is easy to see how this proposed method can lead to the manufacturing savings that all zone based systems share, since they allow for greater tolerances in the tip or end relief areas. This system has the advantage that it will maintain appropriate control over the entire profile or helix.

While the figures and the examples used here were for the profile, it should be realized that the same concepts and advantages apply equally to the helix.

For gears with a modified profile or with a modified helix, the addition of plots showing deviations from the design shape can make understanding the deviations easier.   Use of deviations from design can also make it easier to appropriately control tip, root, or end relief.

1.   ISO  1328-1:2013  Cylindrical  gears  —  ISO  system  of  flank  tolerance  classification  — Part  1: Definitions and allowable values of deviations relevant to flanks of gear teeth
2.   AGMA P109.16, Profile and Longitudinal Corrections on Involute Gears, 1965

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is a senior engineer with Atlas Copco Comptec LLC where he designs gears for high speed integrally geared centrifugal compressors. His career started with the aerodynamic design of compressor impellers, shifted to the design of compressor control systems, and then moved to general research and development of centrifugal compressors. He has been designing gears for over 20 years and, during that time, has also been very active on AGMA committees. He currently serves as the vice chair of the AGMA Gear Accuracy committee, the Computer programming committee, and the Nomenclature committee. Rinaldo is also a member of the Helical Enclosed Drives High Speed Units Committee. He is the United States delegate to ISO TC60/WG2 “Accuracy of gears” working group and is the convener of the dormant ISO TC60/SC1/WG4 Terminology and notation of gears. He also serves on the API 613 taskforce. He has been licensed as a professional engineer in both Wisconsin and New York, has been granted 4 patents, and was the recipient of the AGMA distinguished service award in 2009.