Mathematical model of a straight bevel gear on the straight bevel Coniflex generator and gear flank correction

Coniflex cutting, a popular mass production method for straight bevel gears, employs two giant interlocked circular cutters to generate tooth surfaces. By controlling the tool pressure angle, Coniflex cutting enables profile and lengthwise crownings that result in advantageously low assembly sensibility. Coniflex gears are therefore widely used in the manufacture of a variety of gearboxes. This method is generally used on a Gleason dedicated traditional machine. Although the new CNC machine can implement this method through Gleason software, the technical details of their application are never revealed. Gleason has now stopped producing traditional machines. But the durability, low cost, and ease of setting up these machines have ensured their continued widespread use in the industry. As a result, their users, having no mathematical model of tooth surfaces, have traditionally reduced manufacturing errors by observing the position of the tooth contact bearing and applying a trial-and-error technique. This method is not only time consuming but also results in unpredictable contact performance.

To address these issues, this article proposes a mathematical model of a Coniflex bevel gear produced on a dedicated machine, whose coordinate systems between cutters and work gear are empirically well-defined. This mathematical model can also be used to derive the cutting position of the CNC machine. Once the tooth surface is derived from coordinate transformation and gear enveloping theory, ease off and tooth contact analysis can be conducted numerically; after which flank correction is achieved using sensitivity analysis and optimization methods. Both the proposed model and the flank correction are validated using cutting experiments on the Gleason Number 104 Straight Bevel Coniflex® Generator.

1 Introduction

The gear industry employs several mass production methods for manufacturing the straight bevel gears (SBG) used in gearboxes and automotive differentials, including revacycle broaching, two-tool planning, cutting with dual interlocking circular cutters (DICC), and forging process. The DICC method’s ability to handle a combination of profile and lengthwise crownings and high precision makes it extremely popular. It can only be implemented on a dedicated cradle-type machine. Gleason Works introduced this method used on the PHOENIX^® CNC bevel gear machine. Despite having earned much attention for its high precision and low production costs, the modern CNC machine’s high purchase cost is prohibitive for some manufacturers. The traditional machine has not been replaced by its CNC sibling due to the former’s superior manufacturing performance and low cost. Commercial considerations have discouraged it from revealing the application’s technical details. This lack of a mathematical model for the Coniflex SBG has led to the absence of nominal tooth surface data for measurement and difficulty in conducting either tooth contact analysis (for design) or flank correction (for manufacture). In manufacturing, the machine settings of the gear are fixed, and the operator adjusts those of the pinion for flank correction by observing the position and size of the gear pair’s tooth contact pattern. Only a skilled operator can correct gear errors, and different batches of manufactured gears are not interchangeable.

The available literature has proposed several mathematical models of the SBG tooth, including Al-Daccak et al.’s adoption of a spherical involute to establish the SBG tooth surface [1]. However, this gear can only be produced by an end mill with low productivity and, being perfect, has none of the tooth surface crowning that causes assembly problems in the presence of manufacturing errors. Instead, Ichino et al. [2] used a virtual quasi-complementary crown gear to produce an SBG tooth surface with only profile modification. While Chang and Tsay [3] proposed a mathematical model of the SBG tooth surface with octoid form in which the gear is cut using a two-tool generating method. Shih and her team conducted a series of studies on face hobbing (FH) SBGs over the course of a decade [4-6]. In early 1950, Gleason Works invented the SBG cutting machine and the patented DICC method [7]. These resources include the DICC type disc milling cutters with an internal conical cutting surface on one side to produce lengthwise-crowning teeth [8] and the formulas for tool parameters, blank parameters, and machine settings for the Coniflex machine [9], which refine the blank SBG parameter definitions given in the AGMA gear standard [10]. As earlier emphasized, the company has never provided a mathematical model of the tooth surface. Shih and Hsieh thus used a virtual machine to derive 5-axis nonlinear coordinates for the DICC method [11]. While Alfonso et al. [12] established a mathematical model of the Coniflex SBG based on a crown rack of SBG and then used finite element analysis to investigate the effect of the cutter radius on the gear’s maximum bending stresses. However, neither study provided a mathematical model of the tooth surface produced by the traditional machine. Meaning that the modern correction method cannot be adopted for reducing the errors of Coniflex SBGs.

To facilitate such correction, this article proposes a mathematical model of a tooth surface produced on a straight bevel Coniflex generator that can be used to reduce manufacturing errors via flank correction. Once mathematical models of the cutters and the coordinate systems between the cutters and the work gear on a cradle-type machine are established, the machine settings are converted using data provided by the machine manufacturer [9]. The cutter locus is then derived via coordinate transformation from the coordinate systems of the cutter to those of the work gear. According to the gear-enveloping theory, the tooth surface can be calculated from the equation of meshing and its correctness verified via ease off and tooth contact analysis. The machine corrections needed for flank correction can then be estimated using sensitivity analysis and/or optimization methods. Both the proposed model and the resultant flank corrections are validated using cutting experiments performed on the Gleason No. 104 Coniflex^® generator. The model also can be used to derive the cutting position of new CNC bevel gear cutting machine.

2 Dual Interlocked Circular Cutters

Figure 1a depicts a pair of the tilted interlocked circular cutters used in the DICC method, which are positioned horizontally on the machine. These disc cutters have an internal conical cutting surface on one side that offers a concave tooth to produce SBG with convex teeth. The cutter geometry and its coordinate systems are as illustrated in Figure 1b, with the cutter edge defined in coordinate system S_a and the cutter fixedly connected to coordinate system S_t . The parameters of the cutter blade include profile angle α_b , cutter radius r₀, fillet radius ρ_b , cutter pressure angle ϕ_c , and height of cutter tip h₀ , where the profile angle α_b has a significant influence on the amount of lengthwise crowning on the tooth surfaces.

In general, the cutter edge is a straight line ( r_a^(l)) with a circular arc tip fillet ( r_a^{(f )}), whose position vector is listed in Equation 1. Here, u is a line parameter, and the superscript T stands for transpose.

Where

The fillet center is

Rotating the cutter edge about the y_t – axis (cutter axis) by angle β yields the position vector of the cutter surface, which is represented in coordinate system S_t as follows:

where ± indicates the upper and lower cutters, respectively.

3 Mathematical Model of an SBG Produced by the Straight Bevel Coniflex Generator

Figure 2 depicts a (Gleason) straight bevel Coniflex generator. In addition to a generating mechanism (cradle) for roll ratio R_a, adopts six constant machine settings to move the cutter and work gear positions during the cutting process: (1) cutter offset ∆C, (2) cutter cone distance S_R, (3) space angle ϕ_k, (4) slide base setting ∆B, (5) machine root angle γ_m , and (6) increment of machine center to back ∆A. The first three terms can be separately adjusted for the upper and lower cutters. Additional parameters are the cutter tilted angle ϕ_j  and two machine constants, k_y and k_z. The tilted angle is given dependent on the machine, but the machine constants need to be determined. As Figure 3 shows, coordinate systems S_t and S₁ are fixedly connected to the cutter and work gear, respectively, with auxiliary coordinate systems S_a to S_h indicating the relative motion between them.

The parameters φ_c and φ₁ are the cradle and work gear angles for the generating motion. The locus of the cutting tool in S₁ , expressed as Equation 3, is derived by coordinate transformation from S_t  to S₁:

The transformation matrices are as follows:

where ± indicates the upper and lower cutters. The cradle and work gear angles should satisfy Equation 4 for the generating motion. Substituting Equation 4 into Equation 3 yields locus position r₁ , which consists of only three variables u, β, and φ_c:

According to the gear-enveloping theory, the envelope of the tool locus gives the gear tooth surface, which can be solved from the equation of meshing in Equation 5. This means that the normal vector n₁ is perpendicular to the relative velocity v₁₂ . Hence, using differential geometry, the cross product of the two surface tangent vectors yields the normal vector, and the relative velocity is a time differential of r₁. Each topographic point of the tooth surface has three nonlinear equations — the equation of meshing and two boundary equations of the gear blank — which can be used to solve the three tooth surface variables u, β, and φ_c:

Flank correction is then achieved using two methods: sensitivity analysis and optimization. The first, whose procedures are outlined in reference [14], is used to analyze how a small change (δm_j) in each machine setting (indicated by subscript j) affects the tooth geometry: ∆C, S_R, ∆B, ϕ_k, ∆A, γ_m, and R_a. Since the mathematical model of SBG tooth surface has already been established, the flank normal deviations δQ_ij between the changed tooth surface and theoretical one are determined in Equation 6, which also calculates the sensitivity S_ij  of the machine settings on the tooth surface:

This regression technique yields a correction method that effectively reduces manufacturing errors by enabling derivation of the needed corrections [∆m_j] to the machine settings:

where [S_ij] is the sensitivity matrix built up by the sensitivities S_ij of the machine settings, [∆R_i] is the reverse of the measured errors (corrected target), and subscript i indicates each topographic point on the tooth surface.

The second method chosen for flank correction is optimization (outlined in Figure 4), which begins with derivation of the universal machine settings (all constants) from the given parameters and settings for the Coniflex machine. The theoretical tooth surface is then determined based on the proposed mathematical model, with the nominal data for measurement given by the positions and normal vectors of the tooth surface. If the sum of absolute errors of the work gear is larger than 1,000µm, the machine settings are corrected using the optimization method. These corrections are calculated by finding the numeric minimums of the objective function f, which is the sum of absolute deviations Δ_i between the target ΔR_i and flank deviations ΔQ_i with regard to the corrections Δm_j :

where k indicates the left or right tooth flank, and i is the topographic point from number 1 to n.

4 Derivation of Machine Settings from the Machine Calculating Instructions

Once the cutter parameters, gear blank parameters, machine constants, and machine settings are known — most obtained directly from the machine calculating instructions [9] and the gear blank standard [9] — the tooth surface can be solved. Figure 5 describes calculating the above-mentioned constants k_y and k_z. Where the parameters φ_j and K_D are the inclination angle of the cutter axis and the nominal cutter radius, respectively, and K_E and K_F are the machine constants. The tooth contact length is decided by selecting a pressure angle ϕ_c for the pinion or gear that lies between 22.2° and 17.5°, with blade distance h0 dependent on cutter design.

The initial position of the cutters is the basis for deriving the two constants:

After which the profile angle α_b , space angle ϕ_k and roll ratio R_a are expressed as Equation 10. The parameter ϕ‘_k is the space angle of the machine, and K_B is the machine’s constant. The roll ratio R_a is related to gear parameters such as the pressure angle α_n , pitch angle δ_p, and dedendum angle θ_f .

5 Numerical Examples and Discussion

The numerical example employs a Gleason No. 104 to produce a Coniflex SBG pair (14 × 56) cut by a pair of interlocked cutters with a radius of 114.3 mm and a cutter pressure angle of 17.5°. Figure 6 show the 2D gear blank drawings. This gear pair is illustrated in Figure 7 by a three-dimensional model built using SolidWorks. The machine constants are as listed in Table 1, with some parameters taken from reference [9] and some calculated from Equation 9. The blank parameters, cutter parameters (Table 2), and machine
settings (Table 3) for this traditional machine are calculated using the formulas provided by the manufacturer’s calculation instructions [9] and the gear standard [7]. The universal machine settings for the mathematical model (Table 4) are mostly the same as those for the traditional machine, with only a few needing further conversion based on Equation 10. Substituting all the above parameters into Equations 3 to 5 enables solution of the gear tooth surface, after which the gear pair’s contact performance is evaluated via ease off and tooth contact analysis using the calculation procedures detailed in reference [13]. As Figure 8 shows, the maximum amount of ease off and transmission error is 29.7 µm and 0.11 arc min , respectively.

The profile angle α_b forms an internal cone that induces a lengthwise crowning on the tooth surface for absorbing manufacturing and assembly errors.

6 Cutting Experiments Using the Gleason No. 104 Straight Bevel Coniflex^® Generator

The proposed mathematical models are verified via several gear cutting and corrective experiments in which the Gleason No. 104 bevel gear Coniflex^® machine produces both the pinion and gear (material S45C, HRC 28-32). These experiments use a pair of dual interlocked circular cutters with a nominal diameter of 9 inch (material ASP2023, Hv 905-958) and pinion and gear machining times of 4 min and 15 min , respectively. These nominal data are prepared for a Klingelnberg P40 gear measuring machine. As Figure 9 shows, the resulting sum of absolute errors for the pinion and gear is 2,529µm and 4,272µm respectively with a tooth thickness deviation for the pinion and gear of minus−34µm and minus−103µm respectively.

Because these tooth surface errors are too large for bevel gear production, flank corrections are performed using sensitivity analysis and optimization (rather than the traditional method, which requires a skilled operator). Although corrections are applied to both gears, only the ring gear is illustrated here because of space considerations. As Figure 10 shows, when the machine setting values are varied one by one, the cutter offset ∆C and slide base ∆B affect the spiral angle and tooth thickness; the roll ratio R_a affects the pressure angle and tooth thickness; and the increment of machine center to back ∆A affects the tooth geometry in the bias directions. Although the cutter cone distance S_R, space angle ϕ_k , and machine root angle γ_m have not notably influenced the tooth geometry, the latter two affect the tooth thickness. The machine setting corrections can then be calculated from Equation 7 in accordance with the correction target (Figure 11), which is based on the reverse of the gear errors measured (Figure 9b), and the sensitivity matrix (Figure 12), which is prepared based on the flank sensitivity topographies and Equation 6.

Calculation of the corrections via sensitivity analysis and optimization is done. Because the flank sensitivity topographies reveal that certain settings have a similar influence on the tooth geometry, only four parameters are selected for correction: ϕ_k, ∆C, γ_m, and R_a (see Table 5). An evaluation of the post correction deviations reveals a sum of absolute errors of 525µm for both methods, with simulated topographic errors after correction showing very small differences (see Figure 13). Nonetheless, although both methods have the same precision, optimization takes approximately 94 times longer (optimization 73min 44sec vs sensitivity 47sec), making sensitivity analysis the better choice for flank correction. It is thus the sensitivity analysis-determined corrections that are then added into the original machine settings for a second-round try cut. As Figure 14 shows, just one round of correction results in a reduction of maximum error from 63.3µm to 19.1µm , thickness error from minus−103µm to 27µm, and a dramatic lowering of the sum of absolute errors from 4,271.8µm to 556.2µm. These results, together with the assembled pinion and gear and tooth bearing pattern of the gear pair depicted in Figure 15, confirm that the proposed mathematical model enables the application to a traditional machine of a modern correction technique.

7 Conclusions

Although the Coniflex cutting system is widely applied for SBG production because of its design and manufacture, no mathematical model of its tooth surface has previously been available. This article thus establishes such a model by building coordinate systems between two interlocking cutters and a work gear on a straight bevel Coniflex generator, and then converting the machine settings based on the machine manufacturer’s calculation instructions. Once the mathematical model of the tooth surface is established, evaluations of both gear contact performance (ease off and tooth contact analysis) and individual gear measurements confirm that the proposed mathematical model offers a modern correction technique — applicable to a traditional machine — that drastically cuts production time relative to the conventional method. More specifically, according to the gear cutting and corrective experiments used to verify model correctness, just one round of this flank correction method effectively lowers the maximum error and sum of absolute errors from 63.3µm to 19.1µm and from 4,271.8µm to 556.2µm, respectively. As an additional observation, using both a sensitivity analysis and optimization methods for flank correction identifies the former as the more efficient because of its much lower computing time. The proposed method can be applied to reproduce older parts.

8 Acknowledgments

The authors are grateful to the R.O.C.’s National Science Council for its financial support. Part of this work was performed under Contract No. MOST 107-2221-E011-026. 

9 Nomenclature

k_y, k_z = constants of the mathematical model.

K_B, K_D, K_E, K_F = constants of the Gleason straight bevel Coniflex^®
  machine.

r₀ = cutter radius.

u, β = parameters of the cutting surface.

M_ij = homogeneous transformation matrix from coor-
  dinate system S_j to coordinate system S_i.

n₁ = surface unit normal of the work gear in coordi-
    nate system S₁.

r₁ = locus of the cutting tool in coordinate system S₁.

v₁₂ = relative velocity between the tool and work gear
  represented in coordinate system S₁.

α_b = profile angle of the blade.

∆C, S_R, ϕ_k

∆A, ∆B, gm, R_a = machine settings for the mathematical model.

ϕ_j = angle of inclination of the cutter axis.

φ₁ = rotation angle of the work gear.

φ_c = rotation angle of the cradle.

[∆m_j] = corrections to the coefficients of the universal
  machine settings.

[∆R_i] = correction target.

[S_ij] = sensitivity matrix corresponding to the coeffi-
    cients of the universal machine settings.

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^{Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2022 at the AGMA Fall Technical Meeting. 22FTM10}