Two gear-grinding techniques to improve pitch deviations of ultra-precision gears

Abstract

The ultra-precision master gear with high pitch accuracy has important applications in the calibration of high-precision gear measuring instruments or coordinate measuring machines and in military industries. In order to improve pitch deviations of ultra-precision gear, two gear-grinding techniques to improve pitch deviations of ultra-precision gear were proposed. First, based on the indexing plate system, the working principle of indexing plate and error transferring rules were discussed, and the technical approaches of decreasing the geometric eccentricity were proposed. Then according to the error interference principle, the offset-compensated technique and the neighbour-tooth translocation technique were proposed to improve the pitch machining accuracy of ultra-precision master gear. Finally, precision gear-grinding experiments were conducted by adopting the neighbour-tooth translocation technique based on offset-compensated technique. And the single pitch deviation f_p of the test specimen is reduced from 1.06 to 0.63μm and the total cumulative pitch deviation F_p is reduced from 3.11 to 1.98μm with measurement uncertainty U₉₅ = 0.33μm, which both reach the highest class (class 1 defining in ISO 1328-1:2013). The results show the efficiency and practicability of the two gear-grinding techniques to machine ultra-precision master gear.

Introduction

Gears are one of the most important basic components in modern industries and are widely used in the field of equipment manufacturing, automobile, ship engineering, aerospace, precision instrument and military industry. With the development of industrial modernization, the requirements of gear machining accuracy are exponentially increasing.

Gear pitch accuracy is one of the most important accuracy indexes, of which the single pitch deviation (SPD) f_p and the total cumulative pitch deviation (TPD) F_p are two of the default inspection items defined in gear international standard ISO 1328-1: 2013. ¹ The accuracy requirement of the ultra-precision master gears (UMGs) that are used to calibrate high-precision gear measuring instruments (GMI) or coordinate measuring machines (CMM) for measuring gear is much higher. Due to the lower machining accuracy of gear pitch and effects of gear profile deviations and tooth surface roughness on the measuring uncertainty, the gear-like artefacts are used to calibrate the measuring uncertainty of GMI or CMM. The Italian scholar Maria Pia Sammartini designed and manufactured the gauge block gear-like artefact (GBA; as shown in Figure 1(a)). ² However, the mounting accuracy of the GBA is much lower (f_p = 19μm and F_p = 45μm). Therefore, the GBA is not suitable for calibrating high-precision GMI or CMM. The Japanese scholar Yohan Kondo has developed a multi-ball artefact (MBA; as shown in Figure 1(b)) to calibrate the measuring uncertainty of the instruments.^3–5 Due to the higher machining accuracy (the sphericity is not more than 100 nm) and lower roughness of the ball, the effect of the MBA on the measuring uncertainty of GMI or CMM is much lower. SPD f_p of the MBA can reach about 2 μm and the TPD F_p can reach about 4μm. Such MBA can be taken as a master gear with class 2 accuracy and can meet the calibrating requirement of the high-precision GMI or CMM.

Figure 1.

Gear-like artefacts: (a) the GBA gear-like artefact and (b) the MBA gear-like artefact.

Since the UMG has high pitch accuracy, high profile accuracy and high helix accuracy simultaneously, it can be used to calibrate several indexes of the GMI or CMM. In order to develop the UMG with higher pitch accuracy, pitch machining techniques of UMG are discussed based on the analysis of the working principles and error transferring rules of the indexing system.

Principles and rules

At present, there are mainly two kinds of indexing systems of the high-precision gear grinder; they are the mechanical indexing system and the computerized numerical control indexing system. According to the different indexing components, the mechanical indexing system is classified into the wheel-worm indexing system, the indexing plate (IDP) system and the end-tooth indexing (ETI) system; among these, the indexing accuracy of ETI system is the highest, followed by the IDP system. Although the ETI system has the higher indexing accuracy, considered the complex machining processes and the higher machining costs, the ETI system is not widely used in the manufacturing field. In contrast, the IDP system has wide utilization in the field of machining UMG, gear shaper cutters, gear shaving cutters and so on. Taking the IDP system in gear grinder as an example, the effects on SPD f_p and the TPD F_p of the gear to be ground are analysed.

Working principle of IDP in gear grinder Y7125

The indexing system of the gear grinder (such as Y7125 made in China) with flat-faced grinding wheel is the IDP system. In gear grinder Y7125, the generating system and the indexing system share one and the same generator. One way of the motion outputted from the generator passes by four-level tower shifting, crank and rocker mechanism, rocker and slider mechanism and cam-baffle mechanism to realize the generating motion of the gear grinder. The other way passes by four-level tower shifting, bevel gear pair, flexible axle, indexing worm and Geneva wheel mechanism to drive the indexing gears and indexing cam simultaneously, and to realize the intermittent motion of the IDP. The rough indexing is realized by choosing different indexing gears. Then the indexing cam rotates beyond the maximum peak, and the locating detent draws back under the function of spring force and clutches in the slot of the IDP, which realizes the precision indexing of the IDP. The indexing accuracy of the IDP is mainly determined by the machining accuracy of the slot and the mounting accuracy of the IDP.

The gear to be ground and the IDP are mounted to one and the same grinding mandrel. If there are no eccentricity errors between them, the IDP and the gear to be ground will have the consistent axis of rotation, and the pitch deviations of the gear are only from the manufacturing error e_m of the slots of the IDP in theory. However, the mounting error e_i (the so-called kinematic eccentricity) between the IDP and the grinding mandrel and the mounting error e_g (the so-called geometric eccentricity) between the gear to be ground and the grinding mandrel are inevitable, which are the main error sources of the gear pitch deviations. The working principle of the IDP is shown in Figure 2. In addition, the choice of the finishing occasion, the uniform of the feed quantity and feed occasion, the finishing circle length and the gear-grinding technique also affect the grinding accuracy of the gear pitch.

Figure 2.

Working principle of the IDP.

Transferring rules of pitch deviation

The kinematic eccentricity e_i and the manufacturing error e_m of the slots of the IDP can cause the indexing error of the indexing system. The total indexing error $E_{"}^{\mathrm{i}}(\theta )["]$ of the IDP can be expressed as

$$E_{\mathrm{i}}^{"}(\theta )=\frac{206[e_{\mathrm{i}}\sin (\theta -{\phi }_{\mathrm{i}})+e_{\mathrm{m}}(\theta )]}{R}$$

(1)

where R (mm) is the working radius of the IDP, θ (°) is the phase angle corresponding to the slot of the IDP and ${\phi }_i$ (°) is the phase angle of e_i (μm) to e_m (μm).

Since the gear to be ground and the IDP are mounted to one and the same grinding mandrel, the total angle indexing error of the IDP will transfer to the gear to be ground by the same value if taking no consideration of Abbe error. Therefore, the TPD of the gear to be ground determined by the total indexing error E_i(θ) (μm) of the IDP can be expressed as

$$E_{\mathrm{i}}(\theta )=\frac{206r}{R}[e_{\mathrm{i}}\sin (\theta -{\phi }_{\mathrm{i}})+e_{\mathrm{m}}(\theta )]$$

(2)

where r (mm) is the reference radius of the gear to be ground.

Since the working radius R of the slot of the IDP is definite value (as to gear grinder Y7125, R = 135 mm), the effect of the total angle indexing error of the IDP on the TPD of the gear to be ground takes on the tendency of direct proportion with the reference radius r. The comprehensive geometric error e_g is mainly from the mounting error and the axial float of the gear-grinding mandrel, and the ending beats of the locking face of the gear blank. The TPD E_g(θ) (μm) of the gear to be ground determined by the comprehensive geometric error e_g can be expressed as

$$E_{\mathrm{g}}(\theta )=e_{\mathrm{g}}\sin (\theta -\alpha -{\phi }_{\mathrm{g}})$$

(3)

where α (°) is the pressure angle of the reference circle, θ (°) is the phase angle corresponding to the blank and ${\phi }_{\mathrm{g}}$ (°) is the initial phase angle of e_g (μm).

Therefore, the TPD F_p(θ) (μm) can be obtained by adding equation (2) to equation (3), that is

$$F_{\mathrm{p}}(\theta )=E_{\mathrm{i}}(\theta )+E_{\mathrm{g}}(\theta )$$

(4)

Because the trigonometric function of equations (2) and (3) is of the same frequency (360°), the overlapping result of individual TPD F_p(θ) also has the same frequency. Finally, the effects of the kinematic eccentricity e_i, the comprehensive geometric error e_g and the manufacturing error e_m of the slots on the individual TPD of the IDP on gear blank can be expressed as

$$F_{\mathrm{p}}(\theta )=\lambda \sin {(\theta -\phi }_0)+C$$

(5)

where λ (μm) is the amplitude of the composed vector, ${\phi }_0$ (°) is the initial phase angle of F_p(θ) and C (μm) is the residual constant that is related to the effects of e_i, e_g and e_m.

Expressing the independent variable θ as the function of the number of teeth i, equation (5) can be rewritten as

$$F_{\mathrm{pi}}=\lambda \sin (\frac{360i}{z}-z_0)+C$$

(6)

where z is the number of teeth, i is the integrity from 1 to z, z₀ is the initial the number of teeth and F_pi is the individual TPD.

The TPD F_p is the largest algebraic difference between the individual cumulative pitch deviations, that is

$$F_{\mathrm{p}}=max(F_{\mathrm{pi}})-min(F_{\mathrm{pi}})$$

(7)

The individual SPD f_pi can be expressed as

$$f_{\mathrm{pi}}=F_{\mathrm{pi}}-F_{\mathrm{p}(\mathrm{i}-1)}$$

(8)

The SPD f_p is the maximum absolute value of all the individual SPD, that is

$$f_{\mathrm{p}}=max\left|f_{\mathrm{pi}}\right|$$

(9)

As can be seen from equation (6), the effects of the kinematic eccentricity e_i and the comprehensive geometric error e_g on the individual TPD take on the tendency of basic harmonic wave. The minimum value of λ can be obtained by choosing the right phase angle between e_i and e_g, which decreases the effects of the kinematic eccentricity e_i and the comprehensive mounting error e_g on the individual TPD F_pi. The kinematic eccentricity e_i can only affect the pitch deviations of the gear to be ground; however, the comprehensive geometric error e_g not only affects the gear pitch deviations but also affects the gear profile deviations. Therefore, during the machining of precision gear-grinding, a decrease in the comprehensive geometric error e_g should be considered first, and then the adjusting radial quantity of the IDP to reduce e_i according to the pitch curve of the gear to be ground.

Gear-grinding techniques

Approaches of decreasing the comprehensive geometric error e_g

1. The sliding shaft system was refitted as the antifriction bearing system which includes two multi-ball axle sleeve (MAS). The rotation error of the gear-grinding principal shaft can be reduced from 2 μm to below 0.5μm by choosing the high-grade (above G5) stainless steel ball with 0–3μm magnitude of interference.

2. The radial adjustable gear-grinding mandrel (RGM) with four symmetrical adjusting screws was designed. The angle between the adjusting direction of the screws and the locating surface is 15°, as shown in Figure 3. The radial run out of the working section of the mandrel can be reduced to about 0.5μm by adjusting the four screws of the RGM. The ending beats of the locking face of the RGM to the rotation line can reach about 1μm. The mounting of the gear blank to the RGM adopts MAS with 0–3μm magnitude of interference.

3. In order to improve the mounting accuracy of the gear blank to RGM, the ending beats of the locking faces of the gear and the RGM were measured, and the optimal mounting position was chosen by error compensation method.

Figure 3.

Radial adjustable gear-grinding mandrel (RGM).

The comprehensive geometric error e_g can decrease to 0.5 μm by the adopting the approaches mentioned above. After that, the effects of e_g on F_p are not more than 1 μm and that of e_g on f_p are still less.

Offset-compensated technique

The bigger harmonic component of F_pi of the gear to be ground is mainly from the kinematic eccentricity error e_i of the indexing system, the comprehensive geometric error e_g of the gear blank and the basic harmonic component of e_m. According to equation (5), we can consider the harmonic error of the F_pi of the gear to be ground is all caused by the kinematic eccentricity e_i error of the indexing system. That is to say, the basic harmonic error of the F_pi can be completely compensated by offsetting the IDP. However, it is very difficult to determine the amplitude λ and the initial phase angle ${\phi }_0$ of F_p(θ) directly. Therefore, we solve the amplitude λ and the initial phase angle ${\phi }_0$ by reverse seeking method.

First, a precision gear-grinding experiment is conducted to make the systematic error (e_i, e_g and e_m) reflect to the gear to be ground as far as possible. Then the amplitude λ and the initial phase angle ${\phi }_0$ of the basic harmonic component can be determined by precision measurement and data fitting. It is the so-called offset-compensated technique (OCT) technique that the gear pitch deviation is compensated by radial setting off the IDP according to the magnitude and the phase angle of the basic harmonic component. The detailed operations of OCT are as follows:

1. First, the structure of the IDP is reformed, which includes finishing a radial datum cylindrical surface, enlarging the size of the radial locating cylindrical surface about 1 mm and adding four adjusting screw holes in radial direction. The radial adjustable IDP is shown in Figure 4.

2. Since the machining datum of the radial datum and the IDP’s slots are the same, the radial datum can be used as the initial datum of the radial adjusting of the IDP. A precision gear-grinding experiment is conducted to copy the kinematic eccentricity error to the TPD of the gear to be ground as far as possible.

3. The error curve of F_pi is drawn and the magnitude and the phase angle of the basic harmonic component are fitted by the least square method, and the radial adjusting quantity Δ and adjusting direction (90° lagging to F_pi) are determined according to equation (2).

4. After adjusting the IDP according to the theoretical value Δ and theoretical adjusting direction, another precision gear-grinding experiment is conducted. Due to the adjusting error, it is hard to completely eliminate the basic harmonic component by one-time adjusting operation. Repeating Step 3, the basic harmonic component of the error curve of F_pi can be thoroughly eliminated in theory. Then the radial position of the IDP that corresponds to the smaller TPD F_p is recorded and taken as the mounting datum of the IDP.

Figure 4.

Radial adjustable IDP.

The basic harmonic component of the error curve of F_p can be reduced by adopting OCT that is considered as one of the most effective methods to improve the machining accuracy of the gear TPD. However, the effects of the OCT on improvement of the machining accuracy of the gear single pitch are not obvious.

Neighbour-tooth translocation technique

The gear SPD f_p is one of the most difficult accuracy items to ensue. The error sources of the gear single pitch are mainly from the manufacturing error of the indexing element (e.g. the IDP) and the random error in machining the gear. Although the manufacturing error of the indexing element is the systematic error of the gear to be ground, the machining error of the indexing element shows some random characteristics, which results that the error curve of f_pi also shows some random characteristics. If the error curve is shifted forward or backward at an angle of one tooth, two error curves of f_pi will interfere with each other, which results that some tooth surfaces that have relative machining allowances are reground. According to the error interference principle, some tooth surfaces that have relative machining allowances are reground to achieve the effects of decreasing the gear SPD by controlling the feed quantity or the number of teeth of the gear teeth that participate in regrinding; this is the basic idea of neighbour-tooth translocation technique (NTT). Since the feeding accuracy directly affects the machining accuracy by adopting NTT, the number of teeth that participates in regrinding processes being slightly more than half the total number should be strictly controlled.

According to the closure measuring technique,^6–8 the algebraic sum of the individual SPD f_pi should be 0, that is

$$\sum _i^z{f}_{\mathrm{pi}}=0$$

(10)

Equation (8) shows that the mean of the set of f_pi, which has random characteristic, is expected to be 0. In order to facilitate research, a set of random number of f_pi between ±f_p is supposed as

$$f_{\mathrm{pi}}=2f_{\mathrm{p}}\mathrm{rand}(1,z)-f_{\mathrm{p}}\mathrm{ones}(1,z)$$

(11)

where rand(1, z) expresses a row vector of z random numbers between 0 and 1 and ones(1, z) expresses az-dimensional row vector and all the components are 1.

According to the differences of the translocation directions and the translocation times, the neighbour-tooth translocation can be realized by one-tooth translocation that contains one-tooth left translocation (curve a₁) and one-tooth right translocation (curve b₁), and two-tooth translocation that contains one-tooth left first then two-tooth right (curve a₂) translocation and one-tooth right then two-tooth left translocation (curve b₂).

Taking f_p = 2μm and the number of teeth z = 30 for example, a set of random error curve (defined as the initial error curve f₀) is created by software MATLAB. And the simulation results of different type neighbour-tooth translocation are shown in Figure 5.

Figure 5.

Simulating results of NTT.

The following conclusions can be drawn from Figure 5:

1. Error curves a₁ and b₁ obtained by adopting one-tooth NTT have the same amplitude. However, the phases of them differ at an angle of a tooth. Error curves a₂ and b₂ obtained by adopting two-tooth NTT have no obvious differences both at the amplitudes and the phases.

2. The overall effects by adopting two-tooth NTT are superior to that by adopting one-tooth NTT translocation.

3. No matter which kind of NTT is used, the effects of some teeth whose deviations show the high–low alternation in the initial error curve f₀ (such as the number of teeth 11–15) on the gear individual SPD f_pi are remarkable. However, the effects of some teeth whose deviations show continuous variation in the initial error curve f₀ (such as the number of teeth 2, 9, 16 and 20) on the gear individual SPD f_pi are not remarkable. Therefore, the improved effect of NTT on the gear individual SPD f_pi depends on the proportion of teeth whose deviations show the high–low alternation.

Gear-grinding experiments

The gear grinder with flat-faced grinding wheel Y7125 (made in China) is taken as the grinding equipment, as shown in Figure 6. The rotation error of the RGM can reach 0.5 μm by adopting the approaches mentioned in section ‘Approaches of decreasing the comprehensive geometric error e_g’; the indexing accuracy of the indexing system is $2\prime \prime$ , which is finished by an ETI device and the dressing accuracy of the dresser is 0.2 μm in 30-mm dressing length. In order to reduce the effects of the gear profile deviations on the gear pitch deviations, the optimal forming principle of ultra-precision involute profile is adopted during the gear-grinding experiment. ⁹ The master gear with module m = 3 mm, the number of teeth z = 40, tooth width b = 25 mm, press angle α = 20° and helix angle β = 0° is taken as the specimen, and a precision grinding experiment is performed using appropriate grinding wheel, grinder parameters and ultra-precision grinding method. ¹⁰

Figure 6.

Gear-grinding equipment with flat-faced grinding wheel (Y7125).

Gear-grinding experiment by adopting OCT

First, the approaches of decreasing the comprehensive geometric error e_g mentioned in section ‘Approaches of decreasing the comprehensive geometric error e_g’ are adopted to improve the mounting accuracy of the gear blank. Then the gear-grinding technique of OCT using the high-precision IDP, which is finished by an ETI device, is adopted. Finally, the precision measurements of the individual TPD F_pi and SPD f_pi are obtained using the multi-step method in an ETI table with measurement uncertainty U₉₅ = 0.33μm. ¹¹ The measurement device is shown in Figure 7, and the pitch measurement results of the specimen are shown in Figure 8.

Figure 7.

Measurement device of gear pitch deviation.

Figure 8.

Initial pitch error curve by OCT.

As can be seen from Figure 8, after adopting the high-precision IDP and the gear-grinding technique of OCT, the TPD F_p is reduced to 3.11 μm and SPD f_p is reduced to 1.06μm. The basic harmonic component of the error curve of F_pi can be further decreased by adopting OCT. However, the effects of the OCT on improvement of the machining accuracy of gear SPD f_p are not obvious. In the next, the gear-grinding excrement by adopting the NTT is conducted to further reduce SPD.

Operating processes of NTT

The operating processes of NTT are shown in Figure 9. First, the fine machining of the gear flank is finished with no translocation (the relative position between the IDP and the gear blank is shown as P₀ in Figure 9(a)), and the result of the f_p is obtained by the precision measurement in precision pitch measurement device, as shown in Figure 8. Then according to the accuracy requirement of the gear blank, the one-tooth NTT or the two-tooth NTT is chosen. If the one-tooth NTT is chosen, we take the gear blank rotate to left (the relative position between the IDP and the gear blank is shown as P₁ in Figure 9(b)) or right relative to the IDP at angle of one tooth. If the two-tooth NTT is chosen, based on the one-tooth NTT, we continue to conduct the two-tooth NTT at angle of two tooth in negative direction (the relative position between the IDP and the gear blank is shown as P₂ in Figure 9(c)). The tooth number of the gear that participates in regrinding processes being slightly more than half the total number of gear is strictly controlled in each process. The comparison of the translocation results by adopting the one-tooth NTT and the two-tooth NTT is shown in Figure 10.

Figure 9.

Operating processes of NTT: (a) the relative position between the IDP and the gear blank as no translocation (P₀), (b) the relative position between the IDP and the gear blank as one-tooth translocation (P₁) and (c) the relative position between the IDP and the gear blank as two-tooth translocation (P₂).

Figure 10.

Effects on SPD by adopting NTT: (a) the initial error curve of f_pi, (b) the error curve of f_pi by adopting one-tooth NTT and (c) the error curve of f_pi by two-tooth NTT.

As can be can be seen from Figure 10, the SPD f_p of the specimen is reduced from 1.06 to 0.75μm by adopting one-tooth NTT, and the relative error is reduced by 29.2%; the SPD f_p of the specimen is reduced from 1.06 to 0.63μm by adopting two-tooth NTT, and the relative error is reduced by 40.6%. The experimental results show that the effect of machining accuracy of single pitch is significant by adopting the NTT, especially the two-tooth NTT. However, the machining mechanism of the NTT and quantitative analysis of the effects still need further study in the future.

In theory, adopting the NTT will not change the variation trend of the error curve of the individual TPD F_pi. However, the pitch deviation of the gear teeth that have the relative machining allowance is further reduced by adopting the NTT. Therefore, the TPD F_p will be reduced slightly. The variation trend of the error curve of the individual TPD F_pi is shown in Figure 11 by adopting the NTT.

Figure 11.

Effects on TPD by adopting NTT: (a) the initial error curve of F_pi, (b) the error curve of F_pi by adopting the one-tooth NTT and (c) the error curve of F_pi by adopting the two-tooth NTT.

As can be can be seen from Figure 11, the TPD F_p of the specimen is reduced from 3.11 to 2.26 μm by adopting one-tooth NTT, and the relative error is reduced by 27.3%; the TPD F_p of the specimen is reduced from 3.11 to 1.98μm by adopting two-tooth NTT, and the relative error is reduced by 36.3%. The experimental results show that not F_p but the f_p is reduced by adopting NTT, of which the effects of the former are more significant than that of the latter.

Conclusion

It can be concluded that based on the analysis of the principle and the error transferring rules of the IDP system, two kinds of gear-grinding techniques to improve the machining accuracy of f_p and F_p are proposed, they are the OCT and the NTT. Initially, the article proposed some approaches of decreasing the comprehensive geometric error e_g, which conclude adopting the MAS and designing the RGM.

Then according to the error interference principle, the article proposed the basic idea of OCT, that is, the gear pitch deviation is compensated by radial setting off the IDP according to the magnitude and the phase angle of the basic harmonic component of F_pi. And the detailed operations of OCT were stated.

In addition, another gear-grinding technique NTT was proposed. According to the characteristics of f_pi, taking the error curve of f_pi shifted forward or backward at an angle of one tooth, two error curves of f_pi will interfere with each other, which results that some tooth surfaces that have relative machining allowances are reground.

Finally, precision gear-grinding experiments were conducted by adopting approaches of decreasing the comprehensive geometric error e_g and OCT. The experimental results show that the machining accuracy of gear pitch deviations can reach the highest class. Based on the OCT, another precision gear-grinding experiment was continued by adopting NTT; the machining accuracies of f_p and F_p are further upgraded. Finally, SPD f_p of the test specimen reaches 0.73μm and the TPD F_p of the test specimen reaches 1.98μm with measurement uncertainty U₉₅ = 0.33μm. The experimental results show the efficiency and practicability of the OCT and the NTT to machine UMGs.