Computerized Investigation of Real Tooth Contact Analysis of Face-milled Spiral Bevel Gears

 Abstract: Real tooth contact analysis of spiral bevel gears is based on the original tooth surface grids (OTSG) formed by coordinate measuring machine (CMM). Since the size of OTSG is smaller than the tooth surface, it is sometimes impossible to get full meshing information. Reverse engineering is a way to solve the problem. The basic idea is to expand OTSG to the tooth surface boundary by reversing the manufacturing parameters of the spiral bevel gear drive. Thus a generalized reversing objective is set up for both of the gear and the pinion, which is the summation of deviations of all nodes between OTSG and corresponding computational tooth surface grids (CTSG) expressed by manufacturing parameters. The gear manufacturing parameters are reversed by observing duplex method. The pinion manufacturing parameters are reversed by attempting the meshing behavior taken as input to local synthesis with modified roll motion. The initial meshing behavior is approximately ascertained by discrete tooth contact analysis based on OTSG, and meshing behavior at the mean contact point is figured out by interpolation method for function of transmission errors and contact path. Having reversed the manufacturing parameters, OTSG is expanded to the tooth surface boundary and real tooth contact analysis is conducted. A zero bevel gear drive of an aviation engine was employed to demonstrate the validity of the proposed methodology. The proposed method makes the real tooth contact analysis practical and provides prospect to improve meshing behavior more precisely.


Introduction
Real tooth contact analysis (RTCA) is usually conducted by numerical calculation method based on the original tooth surface grids (OTSG) of spiral bevel gears obtained by coordinate measuring machine (CMM). RTCA is expected to provide more exact information about meshing behavior, to be an alternative quality examination method apart from rolling test machine and to be applied to the manufacturing of the complete interchangeability of the spiral bevel gears. Furthermore, the technique in the reverse of the spiral bevel gears based on OTSG can be used in the closed loop processing of this kind of gear pairs.
To ensure the level of measurement accuracy, OTSG must be smaller than the nominal tooth surface. This sometimes causes the meshing performances resulted from RTCA imperfect and less meaningful. In order to take advantage of RTCA, OTSG must be expanded to the tooth surface boundary by way of reverse engineering that has been found many successful applications in other fields other than in spiral bevel gears.
The tooth surfaces of spiral bevel gears and the corresponding meshing behavior are closely related with manufacturing parameters. So the necessary condition for expanding OTSG to tooth surface boundary is to reverse the manufacturing parameters first. In order to find the adjustment of actual machine-tool settings, Litvin et al. [1][2] measured the coordinates at the nodes of tooth surface grids of an existed tooth flank by CMM. Then he established a set of over-determined equations that described the deviations between the theoretical tooth surface grids and actual tooth surface grids which included the first order adjustment of actual machine-tool settings. By solving the equations, the errors of the actual manufacture parameters away from the theoretical ones were identified. In addition, Some Chinese researchers [3][4][5] proposed similar methodology to ascertain the errors of manufacturing parameters analogy to Litvin's work. Alfonso et al. [9] proposed a computational approach to design face-milled spiral bevel gear drives, which defines the desired topography for the active surfaces of the pinion followed by a numerical derivation of their finishing machine-tool settings through a bound-constrained optimization algorithm. In order to reduce the manufacturing cycle time and maximize the material cut during the rough-cutting operation, Alfonso et al. [10], once again, proposed a novel numerical approach for determination of machine-tool settings for roughing of the pinion by using a spread-blade face-milling cutter. Mu et al. [11] established a mathematical model for the computerized numerically controlled cradle-style pinion generator with the following design parameters: tool parameters, initial machine settings, and polynomial coefficients of the auxiliary tooth surface correction motion,and developed an optimal model to solve the polynomial coefficients. Zhang et al. [12] proposed a technique for computerized simulation and tangency of gears provided with real tooth surfaces. The deviations of real tooth surfaces from the theoretical ones were caused by the distortion of surfaces during the heat treatment and lapping. Yunn et al. [13] proposed a mathematical model to investigate surface deviations of a real cut pinion and gear with respect to the theoretical tooth surfaces and an optimization procedure for finding corrective machine-tool settings to minimize surface deviations of real cut pinion and gear-tooth surfaces. Su et al. [14] individually determined the modified roll coefficients as well as parts of machine-tool settings for the face-milling of spiral bevel gears by applying a method named reverse tooth contact analysis. Artoni [15] calculated the corrections of the machine-tool settings of the pinion to compensate the tooth surface errors away from the theoretical ones caused by both of the pinion and gear in manufacturing process so as to get the same expected contact pattern, transmission errors, and vibration properties. As to the synthesis of mismatch tooth surfaces of face-milled spiral bevel gears, Litvin and Zhang [16] proposed local synthesis to actively control the contact pattern with parabolic transmission errors. Litvin et al. [17] contrasted the pinion tooth surfaces generated by head cutters with straight and circular blades, respectively. After obtaining the stabilized contact pattern with limited magnitude of transmission errors, they proceeded to investigate the contact pressure and tooth root bending stress by employing loaded tooth contact analysis. Wang and Wang [18] utilized radial setting correction to achieve a prescribed fourth-order function of transmission errors and a curved contact path. Fan [19] proposed new algorithms for tooth contact analysis that were integrated into the Gleason CAGE TM 4 Windows Software package to stabilize the simulation process and shorten the computation time. Sheveleva [20] introduced intermediate tangent grids to the pinion and gear, respectively. The meshing performances were found from the field of distance between the pinion and gear as the mating gear pair rotated about their axes of rotation. Simon [21] analyzed the influence of pinion misalignments and tooth spacing error on the contact pattern and kinematics of the gear member. Tsai [22] designed the machine-tool settings on a four-axis milling machine to achieve a predetermined circular-arc contact path. Alves [23] calculated the bending displacement of spiral bevel gears by combining the finite element model and interpolation to speed up the calculation. Mermoz [24] optimized the bearing contact under load by developing an automatic iteration algorithm combining housing deformation and tooth profile design.
This paper provides a set of techniques to reverse the manufacturing parameters of spiral bevel gears departing from OTSG. The mechanism of reversing the manufacturing parameters is applicable to both the gear and the pinion. However, the reversing strategy and design variables for the gear are different from those of the pinion. Based on the reversed manufacturing parameters, the tooth surfaces of the gear and pinion are expanded to their tooth surface boundary. Thus tooth contact analysis is performed to provide the meshing performances of the real tooth surface.

Original tooth surface grid
The original tooth surface grids (OTSG) of spiral bevel gears are formed by coordinate measuring machine (CMM) with 5 points along tooth profile and 9 points along tooth width. Corresponding coordinates of OTSG on the plane passing through the gear axis of rotation, or projection plane, are expressed as follows: where Xo, Ro are the coordinates of OTSG on the projection plane as shown in Fig. 1; xo, yo, zo are the three components of the position vector ro of OTSG on real tooth surface.

Computational tooth surface grid
Computational tooth surface grid (CTSG) is expressed by manufacturing parameters [16]. CTSG is different from OTSG due to manufacturing and measurement errors. To show the difference between OTSG and CTSG node to node, the node on CTSG must correspond to that of OTSG. This can be achieved by solving the following equations: where xc, yc, zc are the three components of the position vector rc of CTSG on computational tooth surface.

Reversing objective
In order to eliminate the difference between CTSG and OTSG, a reversing objective is set up. Rotating CTSG about its axis of rotation until its centre node is coincided with that of OTSG, tooth surface deviation between them is formed. Summing up the field of distances for all nodes on CTSG and OTSG, we can get the optimization objective for reversing manufacturing parameters, or reversing objective for simplicity as below where rc′ represents the position vector of the rotated CTSG on computational tooth surface; x is the vector of design variables; C(x) are used to calculate the instant manufacturing parameters. The manufacturing parameters of the gear member are calculated according to SB card of Gleason Works, and those of the pinion member are calculated by local synthesis with modified roll motion.
Applying optimization method to Eq. (3), CTSG is expected to approach to OTSG, and the manufacturing parameters are then found simultaneously. Expanding CTSG to the tooth surface boundary, the original tooth surface is reconstructed. This provides the prerequisite for real tooth contact analysis.

Design variables for reversing gear tooth surface
The gear member of spiral bevel gears is formatted or generated depending on whether the pitch angle is greater than 70 degrees or not. Here we focus on the generated tooth surface style. In order to increase manufacturing efficiency, the gear member is generated by duplex method. When reversing the manufacturing parameters of the gear, however, the concave side and the convex side can be reversed separately to increase calculation accuracy. Here we first reverse the concave side, and then reverse the convex side. By analyzing the SB card of Gleason Works, we deduce that the design variables for reversing the concave side are the installment angle q2, the outer blade angle α2e and cutter point radius of the outer blade R2e. Since the gear is generated by duplex method, the design variables left to the convex side only include the inner blade angle α2i and the cutter point radius of the inner blade R2i.

Design variables for reversing pinion tooth surface
The pinion tooth surfaces are generated no matter the gear member is generated or not. However, the concave side and the convex of the pinion must be generated separately so as to match the corresponding tooth surfaces of the gear member. So the reversing order of pinion tooth surfaces can be the same as that of the gear member. Usually the pinion tooth surfaces can be determined by local synthesis after reversing the tooth surfaces of the gear member in addition to meshing information at the mean contact point which we will discuss later. Since local synthesis only controls the meshing behavior and its corresponding tooth surfaces at mean contact point and its neighborhood, modified roll motion is sometimes employed to control the meshing performances other than the mean contact point. With the advent of computer numerical control machines (CNC), such as the free form CNC developed by Gleason Works, correction of machine-tool settings provides more alternatives [24]. Here we apply the conventional modified roll motion to control the tooth surfaces. Based on the above analysis we can conclude that the design variables for both of the pinion concave and convex sides must include the position of the mean contact point, the meshing behavior at the mean contact point, and the higher order coefficients of modified roll motion which usually takes the following form: where Δψ1 is the change of the pinion rotational angle with Computerized Investigation of Real Tooth Contact Analysis of Face-milled Spiral Bevel Gears ·5· respect to linear rotational motion in pinion tooth surface generation; Rp is ratio of cutting at the mean contact point; ψcr1 is the rotational angle of the cradle; Ci (i=2,…,6) is the higher order coefficients.

Discrete tooth contact analysis
Local synthesis requires meshing behavior at the mean contact point as its input to form the machine-tool settings in pinion tooth surface generation. This can be done by discrete tooth contact analysis which is carried out according to the following steps: Step 1: The OTSG of the pinion and gear, with the pinion tooth top facing the gear tooth root, are placed as the designed intersection angle in a meshing coordinate system Sm which is rigidly connected to the housing. To find the initial contact point, the pinion OTSG is kept stationary while the gear OTSG is rotated about its axis of rotation to contact the pinion OTSG at a point by applying the tooth surface scanning method [5,20]. The contacted point usually is not coincided with the node of the grids exactly. Its approximate position is calculated by third-order bi-spline interpolation and projected to the projection plane. The instant contact area, which is nearly an ellipse, can also be found with the tooth surface scanning method when the elastic approach is assumed to be 0.00632 mm for milling and 0.00381 mm for grinding at the point.
Step 2: The pinion OTSG is rotated in a constant increment clockwise and counter clockwise, respectively, and the gear OTSG is rotated accordingly to contact with the pinion OTSG until the meshing point exceeds the boundary of OTSG of the pinion or gear.
The above process results in discrete contact points and transmission errors for single tooth pair which needs to be manipulated as follows.

Meshing behavior at the mean contact point
The discrete transmission error points are fitted by a parabola with second derivative m21′. The top point M (Fig.  2a) on the parabola can be treated as the mean contact point. Thus the mean contact point position denoted by XM2 and RM2 on the projection plane of the gear and its tooth surface (Fig. 2b) is approximately ascertained. The major axis of the contact ellipse at the mean contact point denoted by aM is calculated by taking the mean value of those of the two adjacent contact ellipses just in front of and behind the mean contact point. The contact path is fitted by linear regression. This yields the directional angle η2′ on the gear projection plane which is different from the corresponding directional angle η2 on gear tooth surface. Since local synthesis [17] requires η2 as one of its input rather than η2′, a conversion from η2′ to η2 must be made [25].

Numerical example
A zero bevel gear drive in the accessory gear box of a helicopter engine, belonging to spiral bevel gear drives, is used to demonstrate the feasibility of the proposed methodology.

Reversing gear tooth surfaces
The basic parameters of the zero bevel gear drive are shown in Table 1. The blank data of the gear before reversing are calculated and shown in Table 2. The concave and convex sides of the gear are reversed separately as shown in Fig. 3 and Fig. 4. In order to show the deviations between OTSG and CTSG clearly, the field of distance is enlarged by 10 times. The reversed machine-tool settings contrast to the initial ones are listed in Table 3. The blank offset and machine center to back are both taken as zeros. The reconstructed gear concave and convex sides are shown in Fig. 5

Discrete tooth contact analysis (DTCA)
The blank data of the pinion are listed in Table 4. DTCA based on CTSG results in the function of transmission errors and the tooth contact pattern for pinion concave and convex sides as shown in Fig. 6 and Fig. 7, respectively. The small circle "○" on the dot line in Fig. 6a represents the discrete meshing position. The three adjacent motion curves do not intersect with other. This does not imply that tooth impact may occur during tooth pair change but means that the size of CTSG shown in Fig. 1 in dash line is smaller than the nominal tooth surface in solid line. So CTSG is unable not provide full meshing behavior for single tooth pair. Fig. 7 is analogy to Fig. 6. In order to get the meshing information at the mean contact point, data manipulation is conducted as shown in Fig. 8 and Fig. 9 for pinion concave and convex sides, respectively. The motion curve in Fig. 8a is fitted by a parabola in solid line with several discrete points close to the peak to highlight the meshing behavior in this neighborhood. Similar treatment is performed on the motion curve for the convex side as shown in Fig. 9a but close to the valley. The symbol "•" on the fitted parabola is the highest point, and the corresponding symbol "•" on the linear fitted contact path can be taken as the mean contact point as in Fig. 8b and Fig. 9b. The magnitude of the major axis of the instant ellipse at the mean contact point is taken as the average value of those of the instant ellipses just before and after the mean contact point as shown in Fig. 8c and Fig. 9c. The manipulated meshing information is listed in Table 5.

Initial machine-tool settings and higher order coefficients
Initial machine-tool settings as well as the ratio of cutting are calculated by local synthesis when the meshing information is given. Since local synthesis requires η2 as its input, conversion from η2′ to η2 is conducted for pinion concave and convex sides, respectively, as shown in Fig.  10 and Fig. 11, where the red circles are the results of TCA, while the solid line represents the directional angle of DTCA. The resulted initial machine-tool settings with η2 are listed in Table 6. The initial values of the higher order coefficients for modified roll motion are taken as zeros except C1=1.

Pinion tooth surface reversing
Departure from the initial design variables, the pinion concave and convex sides are reversed as shown in Fig. 12 and 13, respectively, in which the solid line is OTSG and the dash line is CTSG. The reversed meshing performances are listed in Table 5. The corresponding machine-tool settings are listed in Table 6. Based on the reversed machine-tool settings, the reconstructed pinion tooth surfaces are shown in Fig. 14.

Real tooth contact analysis
Having reversed the manufacturing parameters of spiral bevel gears, we carried out the tooth contact analysis and got the following results as shown in Fig. 15 and Fig. 16. Fig. 15a shows the meshing behavior of pinion concave side, while Fig. 16 for the pinion convex side. The functions of transmission errors for the concave and convex sides ( Fig. 15a and Fig. 16a) are not exactly symmetrical respect to the mean contact point. Improvements on the functions of transmission errors can be made to both sides especially for the concave side since its functions of transmission errors for adjacent tooth pairs are just intersected. The tooth contact patterns for both sides (Fig. 15c and Fig. 16c) corresponding to the intersections of the function of transmission errors should also be improved since they are near the tooth surface boundaries especially for the concave side which increases the sensitivity of meshing behavior to misalignments.

Conclusions
(1) In order to evaluate the meshing behavior of real tooth surface of spiral bevel gears, CMM tooth surfaces are expanded to the tooth surface boundary by reversing the manufacturing parameters of spiral bevel gears.
(2) The goal of reversing the manufacturing parameters for the gear and pinion are the same, however, the design variable and the reversing strategy for the gear member are different from those of the pinion member.
(3) The real tooth contact analysis is carried out by the conventional method based on the reversed manufacturing parameters and expanded tooth surfaces. The example shows that the zero bevel gear drive needs to be improved both on the function of transmission errors and the tooth contact pattern.
(4) The proposed method can be used to evaluate the meshing behavior of real tooth surfaces, and provides further prospect to improve the meshing quality.

Declaration
Funding Not applicable

Availability of data and materials
The datasets supporting the conclusions of this article are included within the article.