There is a strong and growing demand for skiving cutters, used for the mass production of gears. Compared to the shaping process, the skiving process avoids a brutally interrupted cut, providing new opportunities for carbide tooling. This presents new challenges for cutter manufacturing: pinion cutters may deviate from the involute profile significantly, and the use of diamond wheels makes dressing relatively expensive and time-consuming. We have developed two complementary methods to address these problems for the index generation grinding process.
Firstly: we have developed a grinding path compensation, allowing for errors in the cutter’s pressure angle and crowning, to be addressed, without having to alter the wheel profile. This has proved effective to the point where we can add a crowning of more than 10 micro-meters to a design, purely using path compensation. Secondly: We have developed an in-process flank profile measurement procedure, using an analog ruby probe. This procedure makes use of modern machine axis control, allowing for a nonlinear path to be followed with better than micro-meter accuracy. The procedure calculates the reference geometry for the flank profile, using the nominal wheel profile and grinding path. The deviations of the flank geometry relative to this reference geometry are measured and graphed. For the case where the reference geometry is an involute curve, the measurement result will be directly comparable to a traditional gear measuring machine report.
When setting up to grind a skiving cutter, we can assess whether the ground flanks match the intended design directly, without having to understand complex geometric design details. One can then instantly see what compensation should be applied for the desired outcome, without removing the cutter from the machine. There is a strong synergy between the two approaches, simplifying the otherwise difficult setup for grinding these cutters.
Introduction
There is a significant and growing demand for carbide skiving cutters. These cutters are used for the mass production of gears [1]. Unlike hobs or shaper cutters, skiving cutters are not well suited for making different gears of the same module (diametrical pitch) with an arbitrary number of teeth and/or a large variation in profile shift. Typically, a single cutter design will be produced for manufacturing a particular gear design. Skiving cutters make it practical to cut gears without using a special purpose machine tool, e.g., the gear blank could be turned on a CNC lathe and then cut with a skiving cutter using live tooling.
Relative to hobs, skiving cutters have the advantage that they can be used to cut internal gears, as well as a small gear on an integral shaft with another larger diameter gear close by, which would foul the hob. Relative to shaper cutters, the cutting load on a skiving cutter tooth eases in and out of the cut, providing fewer issues with tear out at the end of the cut, the potential for better surface finish, and the use of the relatively brittle carbide, instead of high-speed steel (HSS). In addition, the machining time with skiving cutters is considerably less than with shaper cutters. The use of carbide provides opportunities for cutting internal gears with the steel in a harder state than is practical with HSS cutters.
To manufacture a skiving cutter to a high standard of accuracy requires careful setup, process control, and very precise wheel dressing. When manufacturing carbide skiving cutters, diamond wheels are typically used, so dressing is slower and more expensive. For this reason, it is important to try and eliminate the need to re-dress a wheel during setup, to bring the flank profile within the specified tolerance. Here we explore the practicality for the in-process measurement of a pinion cutter’s [2] tooth profile, to be used in conjunction with grinding path compensation, to produce high accuracy (DIN-AA [3]) skiving cutters without having to resort to re-dressing a wheel profile to achieve tolerance. We restrict ourselves to the use of index generation grinding (IGG).
1 Manufacturing Pinion Cutters with Index Generation Grinding
Generation methods are based on manufacturing a gear using a cutting tooth (or grinding wheel) that has the profile of a gear tooth. This cutting tooth sweeps through the same path that the meshing gear tooth would, generating the correct profile by mimicking the actual meshing between a pair of gear teeth. Often, for the case of an involute gear, this is a simple rack tooth with flat planes for the active sections of the tooth flanks. Similarly: Hobs and pinion cutters employ generation methods to cut gears. Gears are also commonly ground (after heat treatment) using generation methods. For the IGG method, the grinding wheel follows the meshing contact of a fictitious rack tooth. This is achieved by the grinding wheel reciprocating back and forth, transversely across the rack tooth. To understand the IGG method (see Figure 1), it is useful to visualize the fictitious rack (a thought aid that does not physically exist) in contact with the cutter. This rack, which has a matching helix angle with the cutter, is in perfect mesh with it so the pair has no backlash, i.e., when the cutter rotates, the rack moves in synchronization with it. The grinding wheel is also synchronized to the rack (consider the rack flanks to be inside out for this), and so as the wheel reciprocates back and forth, the rack may move longitudinally (side to side) due to the cutter’s helix angle and, in turn, the cutter will rotate. To make pinion cutters, a flank clearance must be introduced.
This is done by introducing tapered teeth to the fictitious rack; one simple way to do this is to have a linear profile shift along the rack’s transverse direction [4]. There is usually clearance between the grinding wheel and the space between the cutter teeth, so that it only grinds one of the cutter’s flanks (left or right) at a time, and then the wheel is synchronized to the side being ground.
If the active flanks on the rack are flat surfaces, then the wheel profile that grinds them will be straight, and the cutter flank will, therefore, be an involute helicoid [5, 6]. It is conventional to specify this involute helicoid for each flank (left and right) by two parameters: the base diameter and the base helix angle [4, 7]. If one wishes to manufacture a gear with a nominally involute profile, then a shaper cutter will have a profile that deviates very slightly from this, because the intersection of the rake face with the form relieved cutter tooth profile complicates the geometry. However, the profile will be very close to involute, and the deviation may be effectively accounted for by using a small amount of crowning. For a skiving cutter, this deviation can become more pronounced, as it can for more advanced gear and cutter designs. These deviations are often handled by slightly changing the profile of the grinding wheel. If the curvature on the wheel profile, in conjunction with the cutter’s tip clearance angle, is too large, then the flank profile will change with tool life (the reduction in tool length due to repeatedly sharpening back the rake face). To see this, consider how it is a unique property of the involute curve that, upon altering the distance between a pair of gears, the flank geometry does not change. Further, consider the more curved the wheel profile, the more we deviate from involute geometry. Recall that the wheel mimics a rack meshing with the cutter, and (due to the cutter’s tip clearance) that the distance between this rack and the cutter essentially changes with the tool life. So, in distinction to an involute flank, the profile will change with tool life.
Usually, the combination of wheel curvature and tip clearance is arranged so this is not a significant issue.
2 Measuring the Tooth Profile
Here we first consider measuring the tooth profile using a gear measurement machine (GMM). This may be thought of as a specialist form of a coordinate measuring machine, which has a very high accuracy rotary axis, an analog scanning probe and a suite of special software. A GMM is particularly good at measuring pinion cutter flanks to very high accuracy better than a micro-meter. To do this, it scans the flank in the transverse plane nominally following an involute helicoid specified by the base diameter and base helix angle [8]. The probe, however, is deflected in the normal direction relative to the tooth. Again, we can imagine the contact of the probe following the contact between the gear and a rack, but in this case, the pitch diameter is coincident with the base diameter, i.e., a pressure angle of zero and the helix angle equal to the base helix angle. Because, again, the involute rack flank is a flat surface, and because this fictitious rack does not change orientation through the scan, the normal direction to the flank is constant throughout the scan, i.e., the deflection direction in the machine’s reference frame does not change. Implicit in this scheme is the assumption that any deviation (error) in the flank occurs without significantly changing the normal direction. Geometrically, this is a good approximation for small deviations from involute, and for larger deviations, the results will still be deterministic, although with a nonlinear component. Because the direction of the probe deflection is constant throughout the scan, it is a very simple matter to transform it onto the transverse plane, i.e, divide the deflection by the cosine of the base helix angle. Note this flank scan measurement is measuring the flank relative to some arbitrary rotational position (around the cutter’s rotational axis), i.e., the result is independent of any pitch error.
Our in-process measurement extends this classical GMM approach. Rather than follow the nominal involute helicoid, we will follow the nominal flank resulting from the uncompensated CNC grinding paths and the nominal wheel profile. Henceforth, we will refer to this nominal flank as the flank model. To clarify, once a set of grinding paths and a wheel profile is specified, the flank profile is uniquely determined, and to compute this resulting profile, we use the standard mathematical/computational methods developed (with examples) in [5], also see [9, 10] for how these methods can be used to obtain the geometry of a skiving cutter. In the case of a simple involute profile, our path will be the same as the traditional GMM path and measurement. When the nominal flank is not involute, we will still move in the transverse plane, but we calculate the contact point relative to the nominal flank, and the nominal deflection direction may now change slightly throughout the scan. For small deflection amounts, the measurement will still be linear.
We extend the flank model to include a wheel offset (essentially a profile shift) allowing for the cutter to be measured when it is still oversize.
3 Path Compensation
We have developed our own path compensation. It is based upon computing the radius of the point on the wheel (radial wheel point) that touches a given point on the finished tool. Each radial wheel point is projected to where it contacts the tool on the transverse measurement plane. A fixed displacement is assigned to each radial wheel point to adjust the result on the measurement plane to the required outcome. This displacement is then transferred to each position along the grinding path, based on where the radial wheel point is computed to touch the finished tool. In this way, the adjustment on the two- dimensional measurement plane is extended to the three-dimensional grinding path (or tool geometry), i.e., the grinding path is perturbed to mimic what would happen if the wheel profile was adjusted to correct the outcome on the measurement plane. The path compensation relies on there being clearance between the wheel and the cutter flanks, such that when it is grinding the flank on one side, it is clear of the flank on the other side, and thus the left and right flanks can be compensated independently of each other. We have used this compensation extensively on a wide range of different cutter designs ground by different people, and it has proved to be very effective.
4 The Problem
When setting up to grind a cutter with non-involute flanks using the IGG method, the geometry of the flank profile will change with the wheel offset (profile shift), presenting what can be a significant difficulty. Ideally, one would like to measure the flank geometry while the cutter is still oversize. Sometimes this can be accommodated by making the grinding wheel thinner and then approaching the final geometry with an individual rotary offset applied to either flank of the cutter at the final radial position. But this is not always a workable solution, because often it is not desirable (or even feasible) to make the wheel thin enough.
We address this problem by measuring relative to the oversize flank model, providing the opportunity to accurately set up the machine to achieve tolerance before reaching the final size. This is important when setting up to grind an expensive cutter blank with non-involute flanks, allowing the state of the grinding to be assessed while there is still enough material left to correct any accuracy problems.
5 The Experiment
5.1 Experimental Procedure/Specification
Our approach is to grind a cutter oversize using the IGG technique. The oversize is to be achieved by shifting the wheel vertically, away from the cutter, by a fixed amount for all moves (a profile shift). With the IGG technique, for a non-involute profile, the nominal measured tooth profile will change with the profile shift. We have designed our experiment to test our ability to effectively set the compensation for a cutter using in process measurement, when it is still significantly oversize, and then produce a cutter with flank profile in tolerance when the final size is reached. We use a large profile shift to exaggerate the effect.
We designed a skiving cutter that has an essentially involute flank on the left tooth side and significant crowning on the right tooth side. We manufactured this cutter using the IGG technique on a modern CNC tool and cutter grinder using an aluminum oxide wheel and a hardened HSS blank (solid 1” HSS rod).
The wheel was dressed with the CNC control and a dressing roll with the longitudinal position of a freshly dressed reference initially calibrated to high accuracy. From here, no measurement feedback was used for the dressing of the wheel; whatever the machine did was accepted blindly. With the cutter still significantly oversize, the teeth were ground and measured one at a time, and the compensation was adjusted until an acceptable result for the flank profile was obtained at this specific amount oversize. After this, the compensation was not altered, and the cutter was ground down to the finished size. Upon completion, the cutter was also measured using a Klingelnberg P26 GMM, allowing for direct comparison with the in- process measurement.
Different teeth on the cutter were ground in pairs with different profile shifts to provide an opportunity for methodical measurement. Each pair consisted of a compensated and an uncompensated tooth. The most oversize pair of teeth had a shift of 0.24mm, then a pair with 0.12mm and finally a pair on size. For each pair of teeth, we have two measurement sets: the in-process measurement against the flank model and the GMM measurement against the involute reference. The data is conclusive and strongly supports the case we make.
5.2 Cutter Design and Details
We designed the cutter using ESCO PTM 5.0.1.0 software. The cutter itself is a skiving cutter with a 30° shaft angle (right hand helix), no tilt angle, 10° tip clearance angle, a 5° rake angle (30° lead angle stepped rake face), and 20 teeth. This cutter is designed to manufacture (cut) a simple involute 24 tooth spur gear with 20° pressure angle and module 1mm. A simulated rendering of the cutter is provided in Figure 2. The geometrical design of the skiving cutter results in a significant amount of crowning on the right side of the tooth flank (~2.5µm with normal plane deflection), with very little crowning on the left (See Figure 3). The right flank crowning results in non-involute effects upon grinding the cutter. In particular, it causes the profile angle deviation fHα to change with radial infeed under IGG. This change in flank geometry with radial offset makes it challenging to set the cutter up for grinding. The expected scan results for a GMM flank profile scanning measurement at the end of tool (EOT) are shown in Figure 3. We were able to measure at the EOT because we extended the flanks past the EOT during manufacturing.
For this design, we assign an involute helicoid model, for measuring against, with the left flank having base diameter db = 21.348mm, base helix angle βb = 31.114°, the right flank base diameter db = 21.5mm, and base helix angle βb = 25.158°, resulting in the nominal GMM flank scan report shown in Figure 3. The measuring range [11] for both left and right flanks was set from a diameter of 22.2mm to 24.2mm. As we will not be measuring the flank’s lead, the helix angles are not critical, i.e., the results of the flank profile scans will not be sensitive to them.
6 Results and Discussion
6.1 Left of tooth, involute case
The flank to the left of the tooth has very little crowning and is essentially just an involute. While this is the simpler side of the tooth, it allows a direct comparison with the GMM, and the results have some interesting points to understand and discuss. Because the IGG process involves the meshing of gears and because they are involutes for the left flank, we expect the profile to be invariant as a function of the infeed (profile shift). This is so, despite the presence of the flank relief, with the geometry still being an involute helicoid, albeit with altered helix and pressure angles to the gear being cut. Keep in mind it is only approximately involute, with no geometrical circumstance compelling it to be exactly so.
We initially set the machine up and compensated the path with the infeed set 0.24mm oversize (0.48mm on diameter). In process measurement was carried out against the flank model. Results were recorded for a tooth with and without compensation, with a marked improvement for the measured profile resulting from the compensation. The compensation was set with two grinding and then measuring cycles, after which it was not adjusted. These results are shown in Figure 4. Our in-process measurement estimates the profile angle deviation fHα using a linear least squares regression over the evaluation range and computes this in the normal plane (See caption for Figure 4). The GMM computes the difference between the point at the start and end of the evaluation range and reports the fHα value in the transverse plane. To convert from the normal plane to the transverse plane, take the measured value in the normal plane and divide by the cosine of the base helix angle, cos(βb).
The resulting lines of best fit are shown on the graph. Because we do not use the same definition for fHα as the GMM, there can be discrepancies between our results and what is obtained on a GMM. However, experience shows we get good agreement when the value for fHα is very small, which is the objective.
The initial uncompensated grinding result has significant errors for the fHα (-4.6µm) and the crowning Cα (-3.0µm). With the path compensation, these were reduced to -0.1µm and -0.4µm respectively. When setting up the compensation, a tooth can be compensated, then ground at a given infeed, then in-process measured. In this way, at a given infeed, a single tooth at a time can be ground and measured, allowing for the compensation to be tuned by moving onto the next tooth without altering the infeed.
Next, we grind the cutter down to size and then measure. We kept working on different teeth, so the flanks that produced the above measurement could still be measured in the GMM later after completing all grinding, thus the cutter was mounted in the machine one time only, relying on the in-process measurement to achieve the required outcome. We also ground a pair of teeth with the infeed set to 0.12mm, but there is nothing of interest to warrant presenting here. Because the flank is so close to an involute profile, the difference between the flank model and the involute helicoid is very small. In Figure 5, we see a very similar result to what we had with the 0.24mm offset. The largest difference is with the fHα value for the compensated flank, which is now -1.3µm, still well inside the AA tolerance of 2µm [3] and even the informal AAA tolerance of 0.8 × AA = 1.6µm. There are many things that could have caused this small difference, such as wheel wear or thermal drift; however, we did not attempt to investigate this. When we finished all grinding, the cutter was removed from the machine and measured on the GMM. The results on the GMM are shown in Figure 6, which are accurately consistent with the in-process measurement results, within 1µm. The result is very good with the combination of in-process measurement and compensation having considerably improved both the crowning (Cα) and fHα errors to fall within what would be required for a cutter of the tightest tolerances as measured by the GMM. This allows one to reliably set up a cutter for grinding to high accuracy standards without having to remove the cutter from the machine for measurement.
6.2 Right of tooth, non-involute case
The interesting aspect about the non-involute case that we will consider is we expect the flank profile measurement to change with infeed. This presents a challenge when setting up to grind when the cutter blank used for setup cannot be scrapped. While the example used here is non-involute due to the crowning, there are other possibilities. Our approach for in-process measurement is quite general and can handle other types of deviations from an involute profile.
The process used for the left flank was closely followed; in fact, both the left and right flanks were ground together. The initial 0.24mm oversize results are shown in Figure 7. Based on this result, the crowning error on these curves was judged to be acceptable (Cα = -1.6). The fHα value was corrected with compensation resulting in an improved outcome, albeit a very small change. With the benefit of hindsight, it is worth noting these curves differ from the design most strongly near the smallest diameter of the evaluation range, and this is the dominant problem behind the fHα error. This error near the smallest diameter was due to a problem in how we set up the grind paths. The evaluation range was not shifted with the infeed, but the strongest contribution to the fHα error comes from the deviation closest to the small diameter end of the evaluation range. If we assume this error comes from an error in the wheel profile, then shifting the evaluation range with the infeed is the correct course of action. This is supported by a comparison between Figure 7 and Figure 9. In Figure 9, the deviation toward the low diameter has been pushed lower with less of it remaining in the evaluation range. This deviation results in extra material at the lower diameter of the evaluation range, pushing the fHα value smaller, and is largely responsible for the difference in outcome between the two infeed values for the in-process measurement.
Now that we are considering a non-involute flank, it is worth examining the GMM result while still oversize. This allows us to show how the involute-based measurement changes with the infeed and may be seen in Figure 8. Both the amount of crowning (Cα ∼ -5.4µm) and the amount of fHα (∼ -6.6µm) are considerably larger than they were for the in-process measurement shown in Figure 7. The GMM reports its crowning in the transverse plane of the base helix angle, and so in the normal plane this becomes -5.4 × cos(25.16) = -4.9µm. The in-process measurement reports directly in the normal plane and directly against the designed profile. Recall from Figure 3 there is -2.5µm crowning in the design, so we have, -1.6 – 2.5 = -4.1µm. The results between the two measurements compare favorably.
One of the key features arising from the in-process measurement being relative to the flank model is the ability to address the measurement result depending on the infeed. So, we now report a similar set of results to those directly above, but with zero infeed (i.e. the cutter is now on size). The outcome for the in-process measurement is shown in Figure 9 and the GMM in Figure 10. A similar analysis to that given directly above shows the crowning between the two methods to compare very favorably. The crowning in the transverse plane, as taken from Figure 3, should have been about -3.5/cos(25°) = -3.8µm, but according to the GMM report, Figure 10, we ended up with -4.6µm. The in-process measurement accounts for -1.2/cos(25°) = -1.3µm of this leaving -3.3µm, which compares favorably with the expected value of -3.8µm. The results for the fHα measurements are a very good outcome, with favorable comparisons between the in-process and GMM results. The in-process measurement for the non-involute case has proved very successful, allowing the path compensation to be set while the cutter is still oversize, and then a good final result to be achieved without having removed the cutter from the machine to measure during setup.
7 Conclusions
We have tested the combination of path compensation and in-process measurement for controlling the flank profile when setting up the grinding of a pinion cutter using the IGG technique. The in-process measurement is relative to a model of the flank the machine is being instructed to grind (the flank model), as opposed to being specified by an involute helicoid. A notable feature of the flank model is it can readily be used when the cutter is still oversize. We have focused on a systematic process to provide a set of measurements, showing how the approach can be used to obtain the desired result as measured on a GMM. The GMM analyzes the profile angle deviation fHα relative to the two points at the start and end of the evaluation range, while we analyzed it using a least-squares regression. Nonetheless, we were able to provide a clear unambiguous route to the desired geometry. Not only is the in-process measurement suitable for setting up the cutter without having to sacrifice an expensive blank as shown here, it has strong potential for maintaining accuracy during a production run, as well as for use by a small budget business entering the industry that has yet to obtain an expensive GMM.
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