Dr.Qi Fan
Management Summary In addition to the face milling system, the face hobbing process has been developed and widely employed by the gear industry for manufacturing spiral bevel and hypoid gears. However, the mechanism of the face hobbing process is not well known by gear researchers and engineers. This paper presents the generalized theory of the face hobbing process, mathematical models of tooth lengthwise curve generation and tooth surface generation. The face hobbing indexing motion is simulated and visualized. A face hobbing tooth surface generation model is developed, which is directly related to the kinematical mechanisms of a physical bevel gear generator. The developed mathematical models accommodate two categories of the face hobbing system:non-generating (Formate) and generating.
Introduction There are two types of face hobbing processes used to generate the tooth surfaces of spiral bevel and hypoid gears, a non-generated (Formate ®) process and a generated process (Refs. 1, 2, 3). However, the pinion is always manufactured using a generated process. The major differences between the face milling process and the face hobbing process are: (1) in the face hobbing process, a timed continuous indexing is provided, while in the face milling process, the indexing is intermittently provided after cutting each tooth side or slot. Similar to the face milling process, in the face hobbing process, the pinion is cut with a generated method and the gear can be cut with either a non-generated (Formate ®) or a generated method. The Formate method offers higher productivity than the generating method because generating roll is not applied in the Formate method. However, the generating method offers more freedom for controlling tooth surface geometries; (2) the lengthwise tooth curve of face milled bevel gears is a circular arc, while that of face hobbed gears is an extended epicycloid; and (3) face hobbing gear designs use a uniform tooth depth system, while face milling gear designs use a tapered tooth depth system.
Theoretically, the face hobbing process is based on the generalized concept of bevel gear generation in which the mating gear and the pinion can be considered respectively generated by the complementary generating crown gears, as shown in Figure 1.
Figure 1--Basic concept of bevel gear generation.
The tooth surfaces of the generating crown gear are kinematically formed by the traces of the cutting edges of the tool blades, as shown in Figure 2. The generating crown gear can be considered as a special case of a bevel gear with 90° pitch angle. Therefore, a generic term "generating gear" is usually used. The concept of a complementary generating crown gear is considered when the generated mating tooth surfaces of the pinion and the gear are conjugate. In practice, in order to introduce mismatch and crowning of the mating tooth surfaces, generating gears for the pinion and the gear may not be complementarily identical. The rotation of the generating gear is represented by the rotation of the cradle on a hypoid gear generator.
Figure 2--Relationship of the cutting tool, generating gear and the work.
The face hobbing system can use advanced design methodology to formulate the parameters of the generating gears. As a result, face hobbed spiral bevel and hypoid gear sets can offer optimized bearing contact, bias direction and function of transmission errors, resulting in reduced working stresses, vibration and noise. Optimized face hobbed gear sets are less sensitive to errors of alignment.
Related Motions of the Face Hobbing Process In the generated face hobbing method, two sets of related motions are defined. The first set of related motions is the rotation of the tool (cutter head) and the rotation of the work (workpiece), namely,
(1)
Here, ωt and ωw denote the angular velocity of the tool and the work; Nt and Nw denote the number of blade groups and the number of teeth in the work, respectively. This related motion provides indexing between the tool and the work. The indexing relationship can also be represented by the rotation of the tool and the generating gear as,
(2)
where ωc and Nc denote the angular velocity of the generating gear and the tooth number of the generating gear, respectively. Meanwhile, the indexing motion between the tool and the generating gear kinematically forms the tooth surface of the generating gear with an extended epicycloid lengthwise tooth curve, as shown in Figure 3.
Figure 3--Generation of extended epicycloids.
The radii of the rolling circles of the generating gear and the tool are determined respectively by
(3)
and
(4)
here s is the machine radial setting. The second set of related motions is the rotation of the generating gear and rotation of the work. Such a related motion is called rolling or generating motion and is represented as
(5)
where Ra is called the ratio of roll. Basically, the second set of related motions provides generated tooth geometry in the profile direction.
In the non-generating (Formate ®) face hobbing process, only the first set of motion or indexing motion is provided for the gear tooth surface generation. Therefore, the gear tooth surfaces are actually the complementary copy of the generating tooth surfaces.
Tool Geometry
The Gleason face hobbing process uses TRI-AC® or PENTAC® face hobbing cutters.Face hobbing cutters are different from face milling cutters. The cutter heads accommodate blades in groups. Normally, each group of blades consists of an inside finishing blade and an outside finishing blade with reference point M located at a common reference circle, and the blades are evenly spaced on the cutter head (see Fig. 4).
Figure 4--Face hobbing cutter head and blades.
In some specially designed cutter heads, the inside and outside blades are not evenly spaced and therefore the blade radial position is correspondingly adjusted. Basically, the tool geometry can be defined by the major parameters of the blades and their installation on the cutter heads. Since the face hobbing tooth surfaces are kinematically generated by the cutting edges of the blades,an exact description of the cutting edge geometry in space is very important. The blade edge geometry can be described in the coordinate system St that is connected to the cutter head with rotation parameter Θ. The major parameters that affect the cutting edge geometry are:nominal blade pressure angle α, blade offset angle δ, rake angle λ, and effective hook angle κ.
Figure 5 shows a basic geometry of the inside and outside blade edges, which are represented in the coordinate system Sb that is fixed to the front face of the blade. The origin Ob coincides with reference point M at reference height hb. Generally, the blade geometry consists of four sections: (a) tip, (b) Toprem®, (c) profile, and (d) Flankrem.
Figure 5--Blade geometry.
Sections (a) and (d) are circular arcs with radii re and rf respectively. Sections (b) and (c) can be straight lines, circular arcs, or other kinds of curves. Section (c) generates the major working part of a tooth surface. Toprem and Flankrem relieve tooth root and tip surfaces in order to avoid root profile interference and tooth tip edge contact. In order to obtain a continuous tooth surface, the four sections of the blade curves should be in tangency at connections. For a current cutting point P on the blade, the position vector and the unit tangent can be defined in the coordinate system Sb ,
(6)
(7)
where u is the parameter.
Equations 6 and 7 can be represented in the cutter head coordinate system St, as
(8)
(9)
here matrix Mtb denotes the coordinate transformation from Sb to St. Coordinate system Si is used to represent the rotation of the cutter head with an angular displacement Θ. From Figure 4, one can obtain the transformation matrix Mit and represent the blade geometry in system Si as
(10)
(11)
Kinematical Simulation of Face Hobbing Indexing Motion As described above, the indexing motion generates the cutting path or the tooth lengthwise curves that are called the extended epicycloids (see Fig. 3). The indexing motion can be simulated and visualized in terms of the motion relationship between the generating gear and the cutter. Furthermore, such a motion can be represented in the cross-section plane that intersects the blade edges at the reference point M and is perpendicular to the axis of the generating gear so that the motion can be visualized on a plane, as shown in Figure 6.
Figure 6--Simulation of face hobbing indexing motion.
The simulation is based on the following matrix transformation, which represents the relative motion of the cutter head with respect to the generating gear.
(12)
The coordinate system Sg is connected to the generating gear (Fig. 6), and xg and yg are the coordinates of the tooth lengthwise curves, i.e., the extended epicycloids. Coordinates xt and yt specify points on the blades in system St. SH and SV are the initial horizontal and vertical settings of the cutter head. Giving a value to parameter Θ corresponds to a specific cutting position. Figures 6 (a) and (b), respectively, show a single cutting position and multiple cutting positions, from which the kinematical generation of tooth lengthwise curves and the tooth slots can be visualized.
Modeling of Face Hobbing Tooth Surface Generation The face hobbing process can be implemented on the Gleason Phoenix® series CNC hypoid machines (see Fig. 8).
Figure 8--Gleason Phoenix® II 275HC hypoid gear generator.
The machine tool settings and the tooth surface generation model can be derived based on traditional cradle-style mechanical machines. Computer codes have been developed to "translate" the machine settings, and CNC hypoid gear generators move together in a numerically controlled relationship with changes in displacements, velocities, and accelerations to implement the prescribed motions and produce the target tooth surface geometry.
In this paper, a generalized face hobbing kinematical model is developed. The proposed model, shown in Figure 9, is based on the mechanical spiral bevel and hypoid gear generators.
Figure 9--A kinematic model of mechanical hypoid gear generators.
The model consists of eleven motion elements, which are listed in Table 1. The cradle-style represents the generating gear, which provides generating roll motion between the generating gear and the generated work. In the non-generated (Formate®) process, the cradle is held stationary.
Face hobbing machine settings are: (1) ratio of roll Ra; (2) sliding base Xb; (3) radial setting s; (4) offset Em; (5) work head setting Xp; (6) root angle γm; (7) swivel j; and (8) tool tilt i. Although the kinematical model in Figure 9 is based on the cradle-style generators, considering the universal motion ability of CNC machines, the machine settings can be generally represented as
(13)
(14)
(15)
(16)
(17)
(18)
(19)
(20)
The first terms in Equations 13-20 represent the basic constant machine settings and the second terms represent the dynamic changes of machine setting elements, which might be kinematically dependent upon the motion parameters of the cradle rotation angle φ. In current practice, higher order polynomials are used (Ref. 2). These motions can be realized through computer codes on the CNC machine. Equations 13-20 provide strong flexibility for generating all kinds of comprehensively crowned and corrected bevel gear tooth surfaces.
Matrix Representation of Motion Elements In order to mathematically describe the generation process,the relative motions of the machine elements and their relationship have to be represented by a series of coordinate transformations established and rigidly connected to each motion element listed in Table 1. In current pratice, higher polynominals are used (Ref. 2). System Sm is fixed to the machine frame 1 and is considered as the reference of the related motions. System Ss is connected to the sliding base element 11 and represents its translating motion. System Sr is connected to the root angle setting element 10 and represents the machine root angle setting. System Sc is connected to the cradle and represents the cradle rotation with parameter φ. System Sp is connected to the work head sliding setting element 9 and represents the work head setting motion. System So is connected to the work head offset setting sliding element 8 and represents the work head offset setting motion. System Sw is connected to the work 7 and represents the work rotation with angular parameter ψ. System Se is connected to the eccentric setting element and represents the radial setting of the cutter head (see Fig. 11(a)). System Sj is connected to the tilt wedge base element 5, which performs rotation relative to the eccentric element to set the swivel angle j (see Fig. 11 (b)). System Si is connected to the cutter head carrier, which performs rotation relative to the tilt wedge base element to set the tool tilt angle i (see Fig. 11 (c)).
Figure 11--Relationships of coordinate systems Sm, Se, Sj and Si.
During tooth surface generation, at each cutting instant, a point on the blade edge generates a corresponding point on the work tooth surface. Through the coordinate transformation from Si to Sw, the current cutting point P can be represented in the coordinate system Sw, namely,
(21)
(22)
where Mwi (Θ, φ, ψ) is a resultant coordinate transformation matrix with three parameters and is formulated by the multiplication of the following matrices representing the sequential coordinate transformations from Si to Sw,
(23)
where matrices Mwo, Mop, Mpr, Mrs, Msm, Mmc, Mce, Mej and Mji can be obtained directly from Figures 12 and 13 and Equations 21 and 22 can be simply re-written as
(24)
(25)
here subscript "w" denotes that the vectors are represented in the coordinate system Sw.
Figure 12--Pinion and gear tooth surfaces of a face-hobbed hypoid gear drive.
Figure 13--A TCA output interface.
Tooth Surface of a Non-Generated (Formate)Member As discussed above, the non-generated (Formate)gear tooth surface is the complementary copy of the generating gear tooth surface, which is kinematically formed as the cutting path of the blade edge along an extended epicycloid lengthwise curve. In Formate gear generation, the cradle is held stationary and parameter φ is assumed to be zero. Therefore, Equations 24 and 25 can be re-written as
(26)
(27)
which give the position vector and a unit tangent at the current cutting point P on the gear tooth surface. The unit normal of the gear tooth surface can be derived as
(28)
where
(29)
which represents the unit vector of hobbing speed. Equations 26-29 provide equations of position vector, unit tangent, and unit normal of a non-generated gear tooth surface.
Tooth Surface of a Generated Member In addition to the relative hobbing motion or the indexing motion for a generated member, either gear or pinion, generating roll motion is provided and the generated tooth surface is the envelope of the family of the generating surface. The generated tooth surface can be represented as
(30)
where nw is obtained from Equation 28, and the relative generating velocity vw is determined by
(31)
Equation fw(u, Θ, φ)=nwxvw=0 is called the equation of meshing (Ref. 4). Equation 30 defines the position vector, the unit normal, and the unit tangent at the current cutting point P on the generated work tooth surface. All these vectors are considered and represented in the coordinate system Sw that is connected to the work. A unit cradle angular velocity, i.e., ωc=1 can be considered. The equation of meshing is applied for determination of generated tooth surfaces.
Implementation of the Mathematical Models Figures 12 (a) and (b) show geometric models of a pinion and a gear generated based on Equations 26-30. A very important application of the developed mathematical models is the development of tooth contact analysis for the face hobbed spiral bevel and hypoid gears. Tooth contact analysis (TCA) is a computational approach for analyzing the nature and quality of the meshing contact in a pair of gears. The concept of TCA was originally introduced by Gleason Works in the early 1960s as a research tool and applied to spiral bevel and hypoid gears (Ref. 7). Today, TCA theory has been substantially enhanced and generalized and applied in advanced synthesis of face milled spiral bevel gears (Refs. 8, 9).
Based on the developed mathematical models for face hobbing generation of spiral bevel and hypoid gears, an advanced TCA program for the face hobbing process has been developed for both non-generated and generated spiral bevel and hypoid gear drives. The program incorporates simulation of meshing of the whole tooth surface contact including the Toprem and Flankrem contact. Tooth edge contact simulation is also developed and included in the program. Figure 13 shows an example of the output interface of the advanced TCA. Typically, a TCA output illustrates bearing contact patterns on both driving and coast sides of the tooth surfaces and the corresponding transmission errors. Information on the gear drive assembling adjustments is also provided.
Conclusion This paper presents the tooth surface generation theory of the face hobbing process. The kinematics of face hobbing indexing are described and simulated. A generalized spiral bevel and hypoid gear tooth surface generation model for the face hobbing process is developed with the related coordinate systems that are directly and visually associated with the physical motion elements of a bevel gear generator. The generation model covers both Gleason non-generating and generated methods of the face hobbing process. And, it can also be adapted for the face milling process. Based on the developed mathematical model, an advanced TCA is developed and integrated into Gleason CAGE^{TM} for the Windows system, which has been released worldwide.
References
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4. Litvin, F.L. Gear Geometry and Applied Theory, Prentice Hall, 1994.
5. Baxter, M.L. "Effect of Misalignment on Tooth Action of Bevel and Hypoid Gears," ASME paper 61-MD-20, 1961.
6. Krenzer, T.J. "The Effect of the Cutter Radius on Spiral Bevel and Hypoid Tooth Contact Behavior," AGMA Paper No. 129.21, Washington, D.C., October 1976.
7. "Tooth Contact Analysis Formulas and Calculation Procedures," The Gleason Works Publication, SD 3115, April 1964.
8. Litvin, F.L., Q. Fan, A. Fuentes and R.F. Handschuh. "Computerized Design, Generation, Simulation of Meshing and Contact of Face-Milled Formate-Cut Spiral Bevel Gears," NASA Report CR-2001-210894, ARL-CR-467, 2001.
9. Litvin, F.L.and Y. Zhang. "Local Synthesis and Tooth Contact Analysis of Face-Milled Spiral Bevel Gears," NASA Contractor Report 4342, 1991.
Dr. Qi Fan is a bevel gear theoretician at The Gleason Works of Rochester, NY. Prior to that, he worked in the Gear Research Center at the Mechanical and Industrial Engineering Department at the University of Illinois at Chicago. Fan has authored more than 20 papers on bevel gear research.