Optimization of machine tool settings for Spirac hypoid gears by controlling symmetry of contact paths

A novel optimization method to control the symmetry of contact paths on the concave and convex tooth surfaces of the gear then improves the meshing quality was proposed. By modifying the angular setting of the head cutter when cutting the pinion, the direction angles of the two contact paths are equated to estimate their symmetry. The relation between the direction angles is formulated precisely, the influence of the angular setting on the contact paths is investigated, and the equations for obtaining the values of the machine tool settings are derived. The proposed method is applied to a numerical example of a Spirac hypoid gear pair, and the results reveal that the contact paths on the concave and convex tooth surfaces are approximately symmetrical and the transmission errors of both sides are comparable.

hobbing Hypoid gears are widely used to transfer power between two non-intersecting crossed axes, mostly found in the front and the rear axles of all-wheel-drive vehicles or in the rear axles of rear-wheel-drive ones 1,3 .Furthermore, most hypoid gears are manufactured by either the face-milling method or the face-hobbing method [2][3][4] .The face-milling method mainly depends on local syntheses [5][6][7][8] , which predetermine the contact characteristics of the pinion and the gear at a mean point, including the direction of the contact path.Besides determining the contact characteristics around the mean point, Wu et al. 9 presented a theory for the function-oriented design of point contact tooth surfaces.The theory was applied to determine the contact path, major axial length of the instantaneous contact ellipse, and higher-order acceleration as required.Compared with the face-milling method, the face-hobbing method only considers the first-order parameters at the mean point in computing the machine tool settings, such as the position of the reference point, pressure angle, and spiral angle at the reference point; nevertheless, it does not consider the contact path 10 .To guarantee the quality of hypoid gears, various strategies, including higher-order modifications to correct the tooth flank form machining errors 11,12 , ease-off flank modifications for tooth correction and meshing performance improvement [13][14][15] , and multi-objective tooth optimization with modification-based loaded tooth contact analysis to improve the overall meshing quality 16,17 have been proposed.However, most of them are based on Gleason's hypoid generator.
There are three common face-hobbing systems for computing the machine tool settings for hypoid gears, including Klingelnberg's CycloPalloid system, Oerlikon's Spirac and Spiroflex systems, and Gleason's Phoenix system 4,18 .In the Spiroflex system, the pinion and the gear are cut using the generated method; the contact paths on the concave and convex tooth surfaces of the gear are approximately symmetrical, and the instantaneous contact ellipses at the reference point are also symmetrical.However, in the Spirac system, the pinion is cut by the generated method, the gear is cut by the nongenerated method (which enhances production), and the symmetry of the contact paths cannot be guaranteed.Consequently, the contact paths must be modified artificially and repeatedly by correcting the machine tool settings during the tooth contact analysis.The Spirac system is applicable under two conditions for the hypoid gears: when the transmission ratio is greater than or equal to 3 and when the pitch angle of the gear is greater than or equal to 60°1 0 .Generally, the hypoid gears cut by the Spirac system are called Spirac hypoid gears.For Spirac hypoid gears, few studies have been conducted to develop a universal hypoid generator mathematical model 19 , a flank-correction methodology from the six-axis

Theoretical background
In the face-hobbing method, the cutting of tooth surfaces is a continuous indexing process.As shown in Fig. 1, the concave and convex tooth surfaces of a tooth slot are manufactured simultaneously using blade groups 12 .Each blade group contains an inside blade and an outside one, which are placed at the same reference circle on the pitch plane of the head cutter.In the face-hobbing method, the traces of the cutting edges of the head cutter blades form the teeth of virtual generating gear.The curve of the tooth trace of the generating gear forms an extended epicycloid.
In the generated method, two sets of related motions are defined: the relative rotation between the head cutter and the blank, and the rolling (or generating) motion, corresponding to the relative rotation between the virtual generating gear and the blank.In the nongenerated method, which is usually applied to the gear, only the indexing motion is considered.

Generation of the gear tooth surface
When the gear is cut by the nongenerated method, the trace of the blade on the blank is the tooth surface of the gear.Figure 2 shows the relative position between the head cutter and the gear with the right-hand teeth.The reference plane, T, is the basis of the relative position between the gear and the generating gear.P 0 is the pitch point of the blades and its distance to the axis of the head cutter is the nominal cutter radius (r 0 ).T 0 is a plane parallel to plane T and that passes through point P 0 with the addendum modification (h x2 ).T′ is a plane perpendicular to the axis of the head cutter and rotating about point P 0 by tilt angle of the head cutter (χ 2 ).M is the reference point of either the pinion or the gear.In order to accommodate the correct orientation with respect to the cutting motion vector, the effective cutting direction of the blades in the head cutter is not perpendicular to the cutter radius vector.δ 02 is the angle between the cutting direction of the blade and the cutter radius vector.β m2 is the mean spiral angle of the gear at point M. R m2 is the gear cone distance of point M. r m2 is the radius of the reference circle at point M. δ M2 is the gear installment angle.When angle δ M2 equals the gear pitch angle δ 2 , origin O p coincides with the gear pitch cone apex O′ 2 .
Coordinate system S(O; i, j, k) is rigidly connected to the head cutter, where origin O is at the intersection point of plane T′ and the axis of the head cutter, axis i directs to point P 0 , and axis k coincides with the axis of the head cutter.Coordinate system S 0 (O' 0 ; i 0 , j 0 , k 0 ) is an auxiliary coordinate system, where origin O' 0 is at the intersection point of plane T 0 and the axis of the head cutter, axis i 0 directs to point P 0 , and axis k 0 is perpendicular to plane T 0 .Origin O 0 is at the intersection point of plane T and axis k 0 , initially located based on the radial setting (E x2 ) and the angular setting (q 2 ) in coordinate system S p (O p ; i p , j p , k p ). Axis i p passes through point M, and axis k p coincides with axis k 0 .Coordinate system S 2 (O″ 2 ;i 2 , j 2 , k 2 ) is rigidly connected to the gear, where origin O″ 2 is at the center of the reference circle of the gear; axis i 2 passes through point M, axis j 2 coincides with axis j p , and axis k 2 coincides with the axis of the gear and points to the heel.The trace of the cutting edge of the head cutter blade is presented in coordinate system S (rigidly connected to the head cutter) by the vector-parametric equation (fully described in Reference 22 ): Where (R 2 ) is a position vector of an arbitrary point of the trace of the cutting edge; u 2 is the distance from the intersection point of the edge of the blade and plane T′ to an arbitrary point along the edge of the blade; ψ 2 is the angle of rotation of the head cutter; α 2 is the blade angle.The subscript inside the parentheses indicates the number of a body the considered quantity belongs to (index 1 indicates a pinion, index 2 indicates a gear, index 4 indicates a generating gear of pinion).The subscript outside the parentheses indicates a coordinate system in which the considered vector is defined.
The gear tooth surface (Σ 2 ) produced by coordinate transformation from coordinate system S to coordinate system S 2 (connected to the gear) is defined by the following equation (based on Fig. 2) 22 : The coordinate transformations between coordinate systems S, S 0 , S p , S 2 (Fig. 2), are performed as it follows: Where To obtain surface Σ 2 in the generating process, the head cutter is rolled with the work gear, and surface Σ 2 are defined by the following: Where the velocity ratio i 20 in the kinematic scheme of the head cutter for the generation of gear tooth surfaces, based on the ratio of the numbers of blades (z 0 ) and teeth (z 2 ), is: The unit normal vector to surface Σ 2 at point M in coordinate system S 2 is:

Generation of the generating gear tooth surface of pinion
In the generated method, the generation of the pinion is based on the concept of the virtual generating gear, and the pinion tooth surface (Σ 1 ) is generated as an envelope of the virtual generating gear (Σ 4 ). Figure 3 shows the relative position between the head cutter, generating gear of pinion, and pinion with the left-hand teeth.The virtual generating gear is similar to the gear.h x1 is the addendum modification, and χ 1 is the tilt angle of the head cutter.Coordinate systems S and S 0 are established similar to the coordinate systems of the gear shown in Fig. 2. Coordinate system S 4 (O 4 ; i 4 , j 4 , k 4 ) is connected to the virtual generating gear, where origin O 4 is at the center of the pitch circle of the generating gear, axis i 4 passes through point M, axis j 4 coincides with axis j p , and axis k 4 coincides with the axis of the generating gear of pinion.Origin O 0 is at the intersection point of plane T and axis k 0 , initially located based on the radial setting (E x1 ) and the angular setting (q 1 ) in coordinate system S p (O p ; i p , j p , k p ). δ 4 is the pitch angle of the generating gear.δ 40 is the generating gear installment angle.
The movable coordinate system S 4 is rigidly connected with the generating gear of pinion.It is constructed by analogy to system S 4 .The generating gear tooth surface of pinion (Σ 4 ) and the unit normal to surface Here, (R 4 ) is a position vector of an arbitrary point of the trace of the cutting edge, presented in system S by the vector-parametric equation (fully described in Reference 22): Where u 1 is the distance from the intersection point of the edge of the blade and plane T′ to an arbitrary point along the edge of the blade, ψ 1 is the angle of rotation of the head cutter, and α 1 is the blade angle.Subscript 4 inside the parentheses indicates the parameter of the generating gear of the pinion.x 4 , y 4 , and z 4 are Cartesian coordinates representing the position of (r * 4 ) 4 .Matrices M i provide the coordinate transformations from coordinate systems S to S 4 , defined by equations: Where Matrix M 44 defines the relation between the pinion and its generating gear rotating through mesh, and it is described by the following equation (Fig. 3): Each of the tooth surfaces described above covers two families of parameter curves, namely, the u-parameter curve and the ψ-parameter curve.For each point on surface Σ 2 , tangent vector α 2u to the u-parameter curve and tangent vector α 2ψ to the ψ-parameter curve are represented in coordinate system S 2 , as follows (based on Fig. 4): The angle between vector α 2u and vector α 2ψ is determined by The normal curvature and the geodesic torsion of surface Σ 2 for the directions of the u-parameter curve and the ψ-parameter curve are determined and denoted by κ 2u , τ 2u , κ 2ψ and τ 2ψ , respectively: As shown in Figs. 4 and 5, an initial position of the gear is the position in which the reference point M, located in the gear pitch cone generatrix.Angle φ M of the gear rotation about its axis k 2 is measured from initial positions.Vector m 2 (φ M , δ 2 ) is the unit vector of the gear pitch cone generatrix and n 2 (φ M , δ 2 ) is the normal vector of the gear pitch cone surface at point M (Fig. 4).Parameter α n2 is the pressure angle of the gear tooth surface at point M. α 2v is the unit vector for the direction of the tooth trace, and α 2G is the unit vector for the direction of the tooth profile.α 2v is perpendicular to α 2G .α 2v and α 2G are in the tangent planes to surface Σ 2 at point M, and α 2v can be represented in coordinate system S 2 as: Here, (18) The normal curvatures and the geodesic torsion of the generating gear tooth surface Σ 4 of the pinion for the directions of the tooth trace and the tooth profile can be represented and denoted respectively by κ 4v , τ 4v , κ 4G , τ 4G , and γ 4uv , γ 4uψ .The detailed derivations, fully provided in Reference 22, are the same as Eqs.(18-34).
In the generation method, the pinion tooth surface Σ 1 is the envelope of the family of generating surfaces Σ 4 .Based on the meshing theory, the normal curvatures and the geodesic torsion (κ 1v , κ 1G , and τ 1v ) of surface Σ 1 for tangent vectors α 1v and α 1G of the tooth trace and the tooth profile at point M are determined with respect to the curvature relations between the mating surfaces and represented by the following equations: Here, κ 41v , κ 41G , and τ 41v are the induction normal curvatures and the induction geodesic torsion, respectively, and their detailed derivation is provided in Reference 22 .
(30) Figure 6 shows the relative position of the pinion and gear when they are meshing at point M. Coordinate system S 0i (O i ; i 0i , j 0i , k 0i ) (i = 1, 2) is rigidly connected to the machine frame.Origin O i is the foot point of the common perpendicular of the axis of the pinion and the axis of the gear; axis k 0i coincides with the axis of the blank, and axis j 0i coincides with the common perpendicular.Origin O″ 2 is in the center of the pitch circle of the gear at point M. Parameter E is the offset, and parameter Σ is the shaft angle.By coordinate transformation, the gear tooth surface Σ 2 is represented in system S 02 as (based on Fig. 6): Where A m2 is the installment distance of the gear, β m12 is the difference between the spiral angles of the pinion and that of the gear, and δ 1 is the pitch angle of the pinion.
The vectors of the axes of coordinate system S M (M, α 2v , α 2G , n 2 ) are represented in system S 02 as (based on Fig. 7): The angle velocity of the pinion is defined as |ω 1 |= 1 rad•s −1 , and the relative angular velocity is represented in system S 02 by.
where i 12 is the transmission ratio at point M, and.
The relative velocity of the pinion and gear at point M is represented by.In the meshing process, surfaces Σ 1 and Σ 2 are in point contact continuously, and it can be assumed that imaginary tooth surfaces Σ 2 ′ and Σ 1 ′ is obtained from the envelopes of surfaces Σ 1 and Σ 2 , and they are in line contact respectively.α 1v and α 2v are the vectors for the directions of the tooth trace of the pinion and the gear.At point M, α 1v = α 2v .Surfaces Σ 2 and Σ 1 ′, as shown in Fig. 7, are in line contact.Γ 2 is the contact path.The relative positions of Σ 1 and Σ 2 ′are the same as those of Σ 2 and Σ 1 ′.
Tooth surface Σ 1 is in conjugation with surface Σ 2 ′; thus, the unit normal vector of the instantaneous contact line is obtained as.

Here,
Tooth surface Σ 2 is in conjugation with tooth surface Σ 1 ′; thus, the unit normal vector of the instantaneous contact line is obtained as.

Here,
The unit vectors tangent to the contact path at point M are denoted by α 1 and α 2 for the pinion and gear tooth surface, shown as Fig. 8, respectively.The direction angle of α i to α 2v is defined as ν i (i = 1, 2): The area around reference point M meets the geometry condition of the tooth contact along the contact path: reference point M is the common point of both tooth surfaces, and a common normal line for both tooth surfaces exists at reference point M. Accordingly, angle ν i is obtained by the following equations: Figure 8. Position of the direction angle of the contact path.

Influence of the Angular Setting of the Head Cutter on the Direction of the Contact Path
In the Spirac system, the installment of the head cutter is determined by parameters E xi and q i for the gear if i = 2 and for the pinion if i = 1 22 .E xi is corrected to satisfy the requirement of the spiral angle at the reference point, whereas q i is not corrected.To further satisfy the requirement of the direction angle of the contact path, one more parameter is required.Thus, q 1 is chosen, and its influence on the contact path is investigated.
A numerical example is considered to analyze the influence, and Table 1 shows the main design parameters of the gear and the pinion.
Figure 9 shows the result of the contact path on the gear tooth surfaces obtained by correcting the value of q 1 .The dotted lines with " + " signify the contact paths when △q 1 = + 0.0003 rad, the dotted lines with " − " signify the contact paths when △q 1 = − 0.0003 rad, and the solid lines signify the contact paths when the value is not corrected.
In Fig. 9, the positive correction of q 1 increases the angles between the contact path and the tooth trace on the concave and convex tooth surfaces simultaneously, whereas the negative correction decreases the angles.Meanwhile, the positive correction makes the contact paths on the concave and convex tooth surfaces move to the tooth heel simultaneously, whereas the negative correction has the reverse effect.
The result proves the validity of the angular setting of the head cutter q 1 to control the direction of the contact path.Considering its influence on the contact path, q 1 is added to control the symmetry of the contact paths on the concave and convex tooth surfaces of the gear.

Building equations
In the face hobbing method, the concave and convex tooth surfaces of the pinion or the gear are cut simultaneously under a single set of machine tool settings.Equations (1-60), described above, can be applied to evaluate the concave tooth surfaces of the pinion and the gear with the replacement of parameters u 1i , ψ 1i , and α 1i , and u 2e , ψ 2e , and α 2e with the original parameters (u 1 , ψ 1 , and α 1 , and u 2 , ψ 2 , and α 2 ) and the convex tooth surfaces with the replacement of parameters u 1e , ψ 1e , and α 1e , and u 2i , ψ 2i , and α 2i with the original parameters (u 1 , ψ 1 , and α 1 , and u 2 , ψ 2 , and α 2 ).
If the contact paths of the concave and convex tooth surfaces are symmetrical, parameters ν 1e and ν 1i will be related by the following equation: Figure 9. Influence of q4 on the contact path.
The current computation of machine tool settings for the Spirac hypoid gears is based on three conditions at the reference point: (a) the reference point is at the prescribed location, (b) the pressure angle at the reference point is equal to the theoretical value, and (c) the spiral angle at the reference point is equal to the theoretical value.The equations are.
Where r r mjk and r n mjk are the radial and axial positions of the reference point, respectively, on tooth surface Σ j in coordinate system S j ; r m2 is the radius of the reference circle at point M; α m4i and α m4e , and α m2e and α m2i are the normal pressure angles of the concave and convex tooth surfaces of the generating gear of pinion and the gear at the reference point, respectively; α mi and α me are the theoretical values of the normal pressure angles; β m4i and β m4e , and β m2e and β m2i are the spiral angles of the concave and convex tooth surfaces of the generating gear of pinion and the gear at the reference point, respectively; β m2 is the theoretical value of the spiral angle at the reference point.
Equation (61) consists of 16 nonlinear equations, which contain 16 unknowns, u 1k , ψ 1k , u 2k , ψ 2k , E x1 , E x2 , α 1k , α 2k , δ 40 , and δ M2 , for solving for a set of values.To achieve symmetry of the contact paths, angle q 1 is taken as an unknown, and Eq. ( 60) is used for another function.Overall, 17 equations obtained from Eqs. ( 60) and ( 61) are applied to solve for the 17 unknowns to achieve the final values of the machine tool settings.

Analysis of the symmetry of the contact path
Table 1 lists the design parameters for the example hypoid gear pair.By the original method which is currently used for calculation 22 , the values of the machine tool settings are obtained and listed in Tables 2, and 3 lists the results of the conditions.The output of the tooth contact analysis (TCA) is shown in Fig. 10a.
Table 2. Machine tool settings by the original method for the numerical example.By the new method proposed in this paper, the values of the machine tool settings are obtained and listed in Tables 4, and 5 lists the results obtained under the different conditions.The outputs of the TCA are shown in Fig. 10b.

Items
Tables 2 and 4 show that the machine tool settings for the gear are modified and determined to be equal, whereas the machine tool settings for the pinion change.Tables 3 and 5 show that the values obtained by the two methods, related to the position of the reference point, the pressure angle, and the spiral angle, are both equal to the theoretical values.Table 3 shows that the direction angle of the contact paths on the concave tooth surface is not equal to the angle on the convex by the original method.In order to obtain a good meshing performance, they need to be modified artificially and repeatedly by correcting the machine tool settings.However, the direction angles of the contact paths on the concave and convex tooth surfaces of the gear obtained by the proposed method are equal in Table 5.This method is practicable and effective for impacting the meshing performance.
In Fig. 10a, there are significant differences in the tooth contact quality and the motion error between the concave and convex tooth surfaces: (i) both contact paths are not symmetrical; (ii) the convex tooth surface of the gear has a small rotation angle error but a long contact zone, which increases the gear sensitivity; and (iii) although the concave tooth surface has a good contact zone, it has a large rotation angle error.In Fig. 10b, the contact paths on the concave and convex tooth surfaces are approximately symmetrical, the motion errors are approximately equal, and the meshing characteristics are approximate.Thus, the proposed method for controlling the direction angle of the contact path at the reference point is effective, and it can improve the tooth contact quality and the transmission performance.

Conclusion
In this paper, a novel optimization method is presented for determining machine tool settings specific to Spirac hypoid gears, aiming to achieve optimal meshing performance with greater efficiency.
In the existing Spirac system, the pinion is cut by the generated method, the gear is cut by the non-generated method, which provides a higher production, but the meshing performance cannot be predicted and controlled.It needs to be modified artificially and repeatedly by correcting the machine tool settings during the tooth contact analysis.In the proposed method, the symmetry of the contact paths on the concave and convex tooth surfaces of the gear is controlled to achieve optimal meshing performance with greater efficiency.The direction angles of the two contact paths are equated to estimate their symmetry, achieved by modifying the angular setting of the head cutter when cutting the pinion.The results show that the contact paths on the concave and convex tooth surfaces are approximately symmetrical and the transmission errors of both sides are comparable.

Figure 2 .
Figure 2. Relative position between the head cutter and the gear with the right-hand teeth.

3 .
Figure 3. Positions of the head cutter, generating gear, and pinion with the left-hand teeth.

Figure 4 .
Figure 4. Positions of the tangent vectors to the parameter curves, tooth profile curve, and toothtrace curve at point M.

Figure 5 .
Figure 5. Directions of the tangent vectors to the parameter curves, tooth profile curve, and tooth trace curve at M.

Figure 6 .
Figure 6.Coordinate systems connected to the machine frame.

Figure 7 .
Figure 7. Relative positions of Σ 2 and Σ 1 and the coordinate system.

Figure 10 .
Figure 10.TCA outputs with the (a) original and (b) new methods.

Table 1 .
Design parameters for the numerical example.

Table 3 .
Results obtained by the original method for the numerical example.

Table 4 .
Machine tool settings obtained by the proposed method.Results obtained by the proposed method.