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-- UUMMCC --

by Dr. Hermann J. Stadtfeld

Vice President, Research & Development, The Gleason Works

Originated: June, 1994

Revised: October, 1999

The Gleason Works ________________________________________________________________________

1000 University Avenue P.O. Box 22970 Rochester, New York USA 14692-2970

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Flank Modifications in Bevel Gears

Using a Universal Motion Concept - UMC

Dr. Hermann J. Stadtfeld

Vice President Research & Development, The Gleason Works

1. Introduction The development of free-form gear cutting machines was originally based on the idea of designing a machine with a minimal number of freedoms that would allow applicating all the existing bevel gear cutting processes with face cutters. This included single-index applications as well as continuous processes (face milling and face hobbing). The design concept of a six-axis free-form machine met the challenge of today's sophisticated controller and electronic drive technology. A tool can take up any position in space, relative to a workpiece, using only three translatory and three rotational degrees of freedom. This fact fulfilled the basic requirement of a free-form machine, namely to provide the relative movement of a cutter (in space) to a workpiece necessary to produce a bevel gear. The basic design of a free-form bevel gear cutting machine is shown in Figure 1.

The simplicity of the concept in Figure 1 eliminates the complex elements of the traditional gear cutting machine. Except for the swinging base pivoting around the machine center, there is no similarity to the traditional cradle-style machine. Geometrical conditions and kinematic processes of the cradle-style machine are transformed into the motion processes of the free-form machine in such a way that an exactly identical tooth flank geometry is produced. This needs complex transformation calculations [1]. The resulting gear cutting process is completely identical, employing the same process parameters and the same cutting tools. For the convenience of the designer and manufacturer of gearing familiar with the traditional machines, an operator interface for data input into the controller was developed, featuring settings identical to the cradle-style machine. This way, an angle for cutter tilt or eccentric can be entered into the controller, even though these elements are not physically present. Another benefit of this convention is the fact that only about eight numerical values are necessary to exactly define the complex motions of a free-form machine. This indicates the extraordinary

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importance of the so-called "basic (machine) setup data" each of which is relevant in the theory of gearing.

Figure 1: Concept of the free-form bevel gear machine

An extension of the free-form machine, which had proved to be well suited for all currently known processes, was to utilize it further for developing innovative tooth flank corrections. Considering individual axes of the free-form machine showed that seperate axis motions were not appropriate for significant gear geometry improvements. The influence of corrective motion of a specific axis on the tooth form continuously changes during the generating process. This condition made it necessary to activate a combination of motion freedoms at all times, in order to achieve a useful effect on the tooth flank. To avoid having to "cram" the machine controller with three or more polynominals for correction, the relationship between three or more axes was combined to a corrective effect, addressed by

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one higher-order poly-nominal (e. g. radial tilt [2]). This approach, being only partly acceptable with respect to gear theory, lead to a series of problems. As, for the purpose of analysis and optimization, tooth flank surfaces must be generated by calculation, a virtual gear cutting machine concept exists as computer software, modelled after a basic gear cutting machine. This model possesses all the degrees of freedom of a traditional cradle-style gear cutting machine, the motions of the real machine being simulated by computer. This allows the calculation of discrete flank surface points for use in tooth contact analysis, coordinate measurement and finite-element calculation. When corrections are combined, the virtual basic machine - just like the real gear cutting machine - has to activate a combination of several axes according to a mathematical principle with only a single polynominal. Therefore, the object is no longer a mere basic machine without machine constants, where the axes are only theoretically relevant. Auxiliary axes must be defined and formal relationships must be programmed. An idea of correction and optimization cannot be limited to the correction program itself, but must be extended to the flank generation program and the machine controller software, as well. Likewise, any improvements, changes or additions are not limited to the correction program, but also call for an update of the flank generating software and the machine controller software. This can cause logistics problems and has only limited potential for the requirements of gearing theory. 2. Finding a Model for Correction When looking for a suitable model to optimize and correct bevel gear tooth flanks, one can easily come to the conclusion that kinematic processes are involved rather than geometrical ones. All geometrical grades of freedom have been utilized in the traditional gear cutting machine. The question whether there may be other geometrical changes allowing new, still unknown corrections to be realized with the axes of the traditional cradle-style machine, must be answered: no. This is because the traditional cradle-style machine is designed after the coordinate system used for defining gear geometry in which all development calculations are made.

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All geometrically relevant degrees of freedom can be addressed in this coordinate system - which is why the real machine was built after it and remained almost unchanged for more than half a century. This fact indicates that the search for new degrees of freedom, with respect to the known coordinate system, points to a kinematic solution. On the other hand, all geometrical parameters of classical gear cutting theory are transposed by cradle roll into kinematic motions in the free-form machine. When a geometrical setting value of the traditional machine becomes a motion sequence of five to six axes in the free-form machine, a traditional machine's kinematic sequence changes into one overlying another kinematic motion sequence, in a vector space. The vector aspect becomes important, as in the free-form machine, the classical coordinate system is continuously moving, changing its location and orientation. That means that, for instance, a modified movement of the sliding base (helical motion) in the free-form machine is not oriented lengthwise to the machine frame, but is skew in space and changes its direction continuously. It is evident from the above that new correction effects will always be of kinematic nature and that they are already available in a free-form machine, without additional mechanical elements. The last paragraph makes clear that the coordinate system of the free-form machine does not offer a mathematically elegant possibility to calculate corrections relevant to tooth geometry. The solution to this problem is most simple. The best basis of all bevel gear geometry calculation is the basic model shown in Figure 2. The model includes the work axis Z1,2 with the

virtual work gear and the tool representing a tooth of the virtual mating gear, rotating around its axis Y4,5 (generating cradle axis). This fact will remain to be important even in alternative flank

geometries, because the final product will still be a gear rotating around its axis, meshing with a mating gear. Significant correction effects, which are only possible with free-form machines, must be translatory and rotational changes of the coordinate axes of the basic gear cutting machine model during generation of the tooth form.

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Figure 2: Virtual basic gear cutting machine with its setting parameters

Figuratively speaking, this means changing all machine settings of a cradle-style machine during the generating process - which would not have been possible in a traditional machine, without a realistic effort. The calculation effort, however, is small. After converting the basic machine geometry and kinematics into motion coordinates of a free-form machine, it is

possible to manufacture an advanced, optimized tooth form, without additional effort, in the same cutting time and with the same precision as a traditional gear. These considerations and the statements made in the following paragraphs are equally applicable to single-index and continuous processes, for cutting and grinding.

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3. Kinematic Correction Mechanisms The aspects discussed above lead to the so-called Universal Motion Concept (UMC). It means

an extension of the existing calculation methods for gearing by eight kinematic correction mechanisms. Two of these are already available, when considering the modifications of the generating roll (Modified Roll) and of motion in direction of the generating gear axis (Helical Motion). Six further mechanisms are therefore necessary to accomplish basic setting changes in the model machine, depending on a leading parameter. In the superior correction concept, the possibility to change a combination of all setting parameters is most important. When designing the software, fourth-order polynominals were used, to achieve a change of each machine setting, as a function of the roll angle. Any higher-order or trigonometric functions could also be used, but have so far proved not to be necessary. The formulas for UMC motions are shown below. The geometrical meaning of the parameters is evident from Figure 2. S = S0 + S1 · Θ + S2 · Θ2 + S3 · Θ3 + S4 · Θ

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Pi = Pi0 + Pi1 · Θ + Pi2 · Θ2 + Pi3 · Θ3 + Pi4 · Θ4

Pj = Pj0 + Pj1 · Θ + Pj2 · Θ2 + Pj3 · Θ3 + Pj4 · Θ4

Σ = Σ0 + Σ1 · Θ + Σ2 · Θ2 + Σ3 · Θ3 + Σ4 · Θ4

Em = Em0 + Em1 · Θ + Em2 · Θ2 + Em3 · Θ3 + Em4 · Θ4

Xb = Xb0 + Xb1 · Θ + Xb2 · Θ2 + Xb3 · Θ3 + Xb4 · Θ4

Xp = Xp0 + Xp1 · Θ + Xp2 · Θ2 + Xp3 ·Θ3 + Xp4 · Θ4

Ra = Ra0 + Ra1 · Θ + Ra2 · Θ2 + Ra3 · Θ3 + Ra4 · Θ4

where: S... cutter radial setting; Pi... cutter tilt Pj... orientation of tilt; Em... axis offset in the machine Xb... work base setting; Xp... machine center-to-back setting Σ... machine root angle; Ra... ratio of roll Θ... roll position

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One problem in realizing the UMC correction concept was the computer simulation of the cutting process which is necessary before the first advanced corrected gearing is manufactured. For computer tooth contact analysis, finite-element calculation or for providing a master gear for coordinate measurement, the theoretical flank surface is required. This must be described by discrete points and normals in a Cartesian coordinate system.

Figure 3: Configuration generating gear - work gear for solving the gearing law

The most elegant solution for calculating the theoretical flanks is the application of the law of gearing in the basic gear cutting machine model. A point of a defined cutting edge generates a point and a normal on the generating gear flank. The arrangement of the generating gear to the work as well as their ratio (Figure 3) allow the solution of the gearing law (and therefore, a point

and a normal of the work flank is found). The analytical solution of the gearing law is not single-valued. One of the solutions represents a point on an external tooth, another one a point on the mating internal tooth. Two other solutions are either double solutions or an invalid complex

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solution. A single-valued solution of the law of gearing is possible with the vector representation of the gearing problem. In order to generate flank surfaces to be created by means of UMC motions, the vector approach was chosen. In all gear cutting methods widely used today, a limited solution of the gearing law was sufficient. In order to accommodate the additional UMC motions, a completely general formula had to be found for the vector approach. Apart from the rotation of the generating gear Θ = Ω3 and the work Ω2, the rotation of the shaft angle ΩΣ and the translatory movement along the three axes TX, TZ2 and TZ3 must be considered. For the first time in

gearing calculation, a completely universal solution was found, which does not need iteration. The vector (XQ, YQ, ZQ) unambiguously representing the created flank point is:

XQ = - XW · sinΣ - TX · cosΣ · XN + Ra · ( -ΩΣ · YW + VX · YN - VZ2 · sinΣ · XN)

YQ = - YW · sinΣ - TX · cosΣ · YN + Ra · (+ΩΣ· XW + VX · XN + VZ2 · sinΣ · YN)

ZQ = - ZW · sinΣ - TX · cosΣ · ZN + Ra · ( ZW - VZ3 · ZN + VZ2 · cosΣ · ZN)

where XW = XP · ZN - (ZP - TZ3) · YP YW = (ZP - TZ3) · XN - XP · ZN ZW = XP · YN - YP · XN

and (XP, YP, ZP)... generating gear flank point (XN, YN, ZN)... generating gear flank normal (TX, T Z2, TZ3)... shaft offset vector (VX, VZ2, VZ3)... relative velocities of the axis

Σ... shaft angle ΩΣ... angular velocity of the shaft angle

UK=1/Ra... ratio of roll It must be noted that a part of the basic-machine parameters (tilt, orientation and radial setting) do not have any influence on the solution of the gearing law. Their influence comes in at an earlier stage and leads to a variation of the form of the generating gear flank. All other basic-machine parameters (offset, cone distance, set-in, shaft angle and ratio of roll) influence the equations above and change the configuration and the speed between the generating gear and

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the work. The closed solution indicated in this paragraph enables the quick and precise calculation of flank points. It was installed in existing calculation programs instead of the conventional approach and was found to be successful. 4. Theory of the Universal Free-Form Corrections That all corrective calculations can be made in the model of the basic gear cutting machine, familiar to all gearing experts, is quite convenient. Due to the universal motions, however, the model assumes a kinematic character. Compared to the familiar, purely geometrical model, one enters a new dimension. At first glance, the possibilities for changing flank form seem to be infinite, causing the problem of too many affectable parameters. Using an iterative process, trying to set the new kinematic parameters on the basis of a desired ease-off topography would, indeed, not lead to a clear solution. Such an algorithm would quickly show an unstable reaction, as each kinematic degree of freedom, regarded individually, also has an uncontrollable effect on tooth thickness and root line.

Figure 4: Interactive manipulation of the ease-off topography

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Just as for the geometrical corrections of first and second order, it became evident with universal motions that particular, elaborate correction effects are required. Input to the computer monitor is done interactively per mouse. In this way, a standard configuration (Figure 4, top)

becomes an optimized configuration, as shown in the center of Figure 4. The difference between the original ease-off topography and its optimized version is expressed as a fourth-order polynominal. From the first-order coefficients, setting changes for the basic machine are calculated, according to the known geometrical correction mechanisms [3]. A great challenge of mathematical potential are the second-order coefficients. A clever solution was found allowing to separate the components A4, A5 and A6 in Figure 4 into lengthwise curvature, profile

curvature and generation-induced flank torsion [5]. Lengthwise and profile curvatures are geometrical correction effects, whereas generation-induced flank torsion calls for a kinematic correction. The coefficients of third and fourth order were also realized as purely kinematic corrections along the path of contact. A correction software named UMCCORR was established, including all the necessary geometrical and kinematic effects. They are derived in a way allowing independent modification of the ease-offs of coast and drive flanks, i.e. they are compatible to the completing process. From the polynominal shown in Figure 4 (displayed only up to third order), the individual effects are filtered and subsequently laid out as a cumulative correction. Table 1

shows the UMCCORR program structure.

U M C C O R R

Calculation of the ease-off difference function

Separation of the first order for drive and coast flank

Filtration of the second order according to a direction strategy

Transformation of higher orders in direction of roll and profile

Consolidation of drive and coast flank corrections

Allocation of correction effects

Cumulation of geometrical and kinematic corrections

Calculation of updated basic machine settings

Table 1: Structure of UMCCORR optimization software

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It is quite remarkable how the cumulation of the known geometrical and the new kinematic corrections offers nearly unlimited possibilities for flank manipulation. The kinematic correction effects will be discussed in the following. As a "correction vehicle", a kinematic triangular flank generation model is used. Figure 5

shows the initial situation for calculation which at first relates to the center of the flank and is extended to several generating positions, in the second step. Meaningful kinematic effects are the rotation of the tool around the flank normal, the rotation of the tool around the tangent to the flank line and a dislocation of the tool parallel to the tangent to the profile. It is understood that each of these vehicles of correction is associated with a combination of almost all the available kinematic degrees of freedom.

Figure 5: Kinematic triangular correction model

A simple and equally interesting correction effect is the tangential shift of the tool along one of the flank profiles. The underlying idea is shown in Figure 6. A vector kC is calculated for the

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center roll position in the direction of which the tool axis is shifted, in order to influence the opposite flank. By such a shift of the tool, the lower flank in Figure 6 remains unaffected. An equidistant stock allowance develops on the top flank in Figure 6, becoming effective along the actual line of contact. The only side effect is a change of the root line by an amount approximating that of the correction. As the corrections take place in the range of hundredths of a millimeter (thousands of an inch), this change of the root line is always neglectably small.

Figure 6: Function of the kinematic correction effect SFCOR

The vector kC is calculated from the rotations (α, β, γ) which are necessary to provide it with the direction of the tangent to the profile in the initial position, Y' in Figure 5. The equation for kC

in Figure 6 shows only the spiral angle β to be a parameter depending on roll. To make

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SFCOR function more perfectly, further roll positions may be observed which are used to address coefficients of function fβ(Θ). The correction as such is effected by addressing the polynominal coefficients C1 thru C4 of

function f(Θ). The tool is shifted during the generating roll on the vector by the amount kC · fβ(Θ)

· f(Θ). Therefore, the tool position vector EX changes to:

EX,kin = EX + kC · fβ(Θ) · f(Θ)

and the amount of correction on the mating flank is derived from: Correction amount = f(Θ)· sin (α1 + α2)

= [C1 · Θ + C2 · Θ2 + C3 · Θ3 + C4 · Θ4] · sin (α1 + α2)

Figure 7: Results of sample calculations with the SFCOR correction effect

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Tests have shown that the SFCOR correction is perfectly suitable for the completing process. Figure 7 shows the application of the corrections on a conjugated specimen gear where both

ease-offs are "zero surfaces" (Figure 7, top). The geometrical basic machine set-up for all four rows shown is identical, only the polynominal coefficients of the eight "basic settings" have been assigned differently. The second row in Figure 7 emerged with correction address no. 2 (for concave flank of pinion) and assigning the first-order coefficient C1. Using the same correction address, only coefficient C2 was assigned in the third row and only coefficient C3 in

the fourth row. Comparison of the coast flank ease-offs proves that, even when drive flank changes are enormous, the distorting influences on the non-corrected flank are completely neglectable or not even present. Because manipulations of the convex and the concave flanks are independent of each other, any parts of the corrections shown in Figure 7 may be realized on both flanks. For practical purposes, the manipulation of flank surfaces with SFCOR offers undreamed-of possibilities never before realizable. The contact line curvature mentioned in the "Handbook of Bevel and Hypoid Gears" [5], which was named "Litvin correction" after its discoverer, Faydor L. Litvin, could only be achieved, so far, with the older rather uneconomical five-cut method. As this correction is based on a change of cutter radius between 5 and 10 mm (¼ and ½ inch), its realization with standard cutting tools was only possible, in special cases. The correction program UMCCORR allows the complete conversion of an existing lengthwise curvature into a contact-line curvature. In gears cut by single-index completing reaching the pressure angle limits with cutter tilt, before the contact pattern has its desired length, this correction can be used for lengthening the pattern. It can also be the basis of an elaborate optimization process for sophisticated gear geometry. An example for the conversion of a pure lengthwise curvature into a pure Litvin correction is shown in Figure 8.

By interactive input (Figure 4), the ease-off at the entrance and the exit points are suppressed to zero. The correction program will then calculate assignment of the polynominal coefficients C1 thru C4, turning the ease-off shown in Figure 8, top into that shown in the center.

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The proof of successful conversion is in the transmission error plotted at the bottom of Figure 8. The contact line curvature does not have any effect in the direction of the path of contact and

therefore does not display any transmission error.

Figure 8: Conversion of lengthwise curvature into a pure contact line curvature

The SFCOR properties allow any equidistant correction along the generating path. Superimposing tool rotation relative to the specific point of the generated tooth trace (FTCOR) on this flank correction even allows generation of a cycloidal tooth form. For the first time in the history of gear cutting, this allows true form-grinding using single indexing, of gears cut by the continuous method.

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5. Independent Optimization of Different Flank Areas In order to provide the universal motion concept for free-form gear cutting machines with sufficient degrees of freedom for future optimization ideas, each flank can be divided into three separate areas. The kinematic basic settings of each area were defined separately, so they can have different values. This way, each of the three can be optimized separately. The only limitation is a smooth transition between these areas to ensure impact-free rolling of the gears.

Figure 9: Approach correction and ENDREM® by three different correction areas

A practical example, how this property can be used successfully, is shown in Figure 9. The

ease-off topographies of a conveniently designed gear pair are shown in Figure 9, top. Without changing the central flank area, an entrance correction at the heel could be applied on the coast side (Figure 9, bottom left) and on the drive side, a considerable end relief (ENDREM®) could

be made (Figure 9, bottom right). By dividing the flank into three areas, the fourth-order polynominals can be used with three different assigments. This does not only add the convenience of utilizing a considerably higher function, e.g. of twelfth order, but also eliminates

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instabilities causing vibration. An eigth-order polynominal already possesses five inflection points causing three and a half harmonic waves in the µm (.000040 inch) range. The periodicity

of this unintended harmonic modification may lead to an audible stimulation with 3.5 times the tooth mesh frequency. 6. Summary The potential of free-form machines was recognized comparatively early. The basic fact, however, that the six degrees of freedom available could be used to obtain any desired movement in space was not yet a conclusive basis for flank corrections and optimization. Compared to the optimization efforts "of the first hour", the idea discussed in this paper is a milestone in gear correction mathematics, just as the free-form machine necessary to put these ideas into use was a milestone in machine-tool building. The starting point is a basic gear cutting machine following the classical design, all the settings of which are changeable depending on generation angle ("the living settings"). As each setting parameter has a significance for gear theory, the variation of these values during the generating process is also significant. Like the geometrical correction ideas which always need a combination of several machine setting values to produce a "pure correction effect", e.g. change of the pressure angle, the kinematic correction effects are a combination of several motions. To date, a total of three combined correction effects have been derived and programmed. The first effect, SFCOR was completely explained already. The second one, FTCOR rotates the work gear and the tool relative to each other around the tangent to the flank in the specific point of the generated tooth trace (Figure 5). The third effect is named FNCOR. It allows a relative rotation between work gear and tool around the flank normal which can be utilized as end relief (ENDREM®).

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The kinematically optimized gear with a flank form derived from minimizing the meshing thrust is described in detail in the "Handbook of Bevel and Hypoid Gears" [5]. Figure 10 shows in the

top section how the ease-off topography can be "put together" using only the two elements of contact-line curvature and of higher-order lengthwise curvature. When trying to realize such optimized geometries, one soon reached the limitations of gear cutting machinery or correction calculation methods. The compromise was the implementation of a multiple-cut method. By means of UMC, kinematically optimized geometries according to Figure 10 can be realized with the completing process, without compromise. The necessary basic elements were presented in Figure 8.

Figure 10: Strategy of kinematic optimization of a bevel gear set

There is almost no limitation to finding new flank optimization ideas, in the future. The goal of the UMC correction concept was to create the calculation basis for bevel gear experts' future work. The success evident from the first development stages show that this future has already begun. 7. Literature

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[1] Goldrich, R. N. Theory of Six Axes CNC Generation of Spiral Bevel and Hypoid Gears

AGMA Fall Technical Meeting, Pittsburgh, Nov. 1989

[2] Krenzer, T. J. The Application of Flared Cup Grinding to new Bevel Gear Tooth

Geometry, CMET, 3rd World Congress of Gearing and Power

Transmission, Paris, Feb. 1992

[3] Stadtfeld, H. J. A Closed and Fast Solution Formulation for Practice oriented Optimi-

zation of Real Spiral Bevel and Hypoid Gear Flank Geometry

AGMA Fall Technical Meeting, Toronto, Oct. 30, 1990

[4] Knaden, M. Optimization of Bevel Gears by Variation of the Gear Machine Settings

MSc Thesis, University of Aachen, 1985

[5] Stadtfeld, H. J. Handbook of Bevel and Hypoid Gears - Calculation, Manufac-

turing, Optimization

Rochester Institute of Technology, Rochester, New York, 1993

Many of the described techniques and machines, as well as many applications of special cycles and methods are protected worldwide by patents or patent applications of The Gleason Works. The mentioned registered trademarks® of The Gleason Works are marked as such.

The Gleason Works

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