Patent Publication Number: US-6712566-B2

Title: Machine and method for producing bevel gears

Description:
This application claims the benefit of U.S. Provisional Application No. 60/269,328 filed Feb. 16, 2001. 
    
    
     FIELD OF THE INVENTION 
     The present invention is directed to gear manufacturing machines and more particularly to machines for cutting and grinding bevel gears. 
     BACKGROUND OF THE INVENTION 
     In the production of gears, especially bevel gears, two types of processes are commonly employed, generating processes and non-generating processes. 
     Generating processes can be divided into two categories, face milling (intermittent indexing) and face hobbing (continuous indexing). In generating face milling processes, a rotating tool is fed into the workpiece to a predetermined depth. Once this depth is reached, the tool and workpiece are then rolled together in a predetermined relative rolling motion, known as the generating roll, as though the workpiece were rotating in mesh with a theoretical generating gear, the teeth of the theoretical generating gear being represented by the stock removing surfaces of the tool. The profile shape of the tooth is formed by relative motion of the tool and workpiece during the generating roll. 
     In generating face hobbing processes, the tool and workpiece rotate in a timed relationship and the tool is fed to depth thereby forming all tooth slots in a single plunge of the tool. After full depth is reached, the generating roll is commenced. 
     Non-generating processes, either intermittent indexing or continuous indexing, are those in which the profile shape of a tooth on a workpiece is produced directly from the profile shape on the tool. The tool is fed into the workpiece and the profile shape on the tool is imparted to the workpiece. While no generating roll is employed, the concept of a theoretical generating gear in the form of a theoretical “crown gear” is applicable in non-generating processes. The crown gear is that theoretical gear whose tooth surfaces are complementary with the tooth surfaces of the workpiece in non-generating processes. Therefore, the cutting blades on the tool represent the teeth of the theoretical crown gear when forming the tooth surfaces on the non-generated workpiece. 
     Conventional mechanical gear generating machines for producing bevel gears comprise a work support mechanism and a cradle mechanism. During a generating process, the cradle carries a circular tool along a circular path about an axis known as the cradle axis. The cradle represents the body of the theoretical generating gear and the cradle axis corresponds to the axis of the theoretical generating gear. The tool represents one or more teeth on the generating gear. The work support orients a workpiece relative to the cradle and rotates it at a specified ratio to the cradle rotation. Traditionally, conventional mechanical cradle-style bevel gear generating machines are usually equipped with a series of linear and angular scales (i.e. settings) which assist the operator in accurately locating the various machine components in their proper positions. 
     It is common in many types of conventional mechanical cradle-style bevel gear generating machines to include an adjustable mechanism which enables tilting of the cutter spindle, and hence, the cutting tool axis, relative to the axis of the cradle (i.e. the cutter axis is not parallel to the cradle axis). Known as “cutter tilt,” the adjustment is usually utilized in order to match the cutting tool pressure angle to the pressure angle of the workpiece, and/or to position the cutting surfaces of the tool to appropriately represent the tooth surfaces of the theoretical generating gear. In some types of conventional mechanical cradle-style bevel gear generating machines without a cutter tilt mechanism, the effects of cutter tilt may be achieved by an altering of the relative rolling relationship between the cradle and workpiece. This altering is also known as “modified roll.” 
     In the recent past, gear producing machines have been developed which reduce the number of machine settings necessary to orient a tool relative to a workpiece. These machines replace some or all of the settings and movements of the conventional mechanical cradle-style machine with a system of linear, rotational, and/or pivoting axes. 
     SUMMARY OF THE INVENTION 
     The present invention is directed to a machine for manufacturing bevel and hypoid gears comprising a machine column including a first side and a second side. A first spindle is movably secured to the first side with the first spindle being rotatable about a first axis. A second spindle is movably secured to the second side with the second spindle being rotatable about a second axis. The first and second spindles are movable linearly with respect to one another in up to three linear directions with at least one of the first and second spindles being angularly movable with respect to its respective side. The angular movement of at least one of the first and second spindles being about a respective pivot axis extending generally parallel with its respective side. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is an isometric view of a first embodiment of the inventive gear manufacturing machine with the tool and workpiece disengaged. 
     FIG. 2 is an isometric view of the first embodiment of the inventive gear manufacturing machine showing a cutting tool engaged with a pinion. 
     FIG. 3 is a top view of the gear manufacturing machine of FIG.  2 . 
     FIG. 4 is an isometric view of the first embodiment of the inventive gear manufacturing machine showing a cutting tool engaged with a ring gear. 
     FIG. 5 is a top view of the gear manufacturing machine of FIG.  4 . 
     FIG. 6 illustrates a conventional mechanical cradle-style bevel gear generating machine with cutter tilt. 
     FIG. 7 is a schematic top view of a conventional mechanical cradle-style bevel gear generator. 
     FIG. 8 is a schematic front view of a conventional mechanical cradle-style bevel gear generator. 
     FIG. 9 is a side view of the tool in FIG.  8 . 
     FIG. 10 is a schematic top view of the cutting tool and workpiece of the first embodiment of the present invention. 
     FIG. 11 is a view along the cutting tool axis of FIG.  10 . 
     FIG. 12 illustrates pivot axis F referenced in a coordinate system based on the reference plane of the cutting tool in the first embodiment of the present invention. 
     FIG. 13 shows the coordinate system of FIG.  12  and the coordinate system of the first embodiment of the inventive machine. 
     FIG. 14 shows the coordinate systems of the cutting tool, X c -Z c , and the inventive machine, X-Z, in the first embodiment of the present invention. 
     FIG. 15 is a machine axes motion diagram for a pinion cut on the machine embodiment shown in FIGS. 1-3. 
     FIG. 16 illustrates a pivot axis placement associated with a workpiece spindle. 
     FIG. 17 exemplifies an alternative form of the machine column. 
     FIG. 18 depicts vertical machine motion being associated with a tool spindle. 
     FIG. 19 is a top view showing pivot mechanisms being included with both tool and workpiece spindles. 
     FIG. 20 illustrates horizontal guides being located inward of vertical guides for movement of a workpiece spindle. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     The details of the present invention will now be discussed with reference to the accompanying drawings which illustrate the present invention by way of example only. In the drawings, similar features or components will be referred to by like reference numbers. 
     In the context of the present invention, the term “bevel” gears is understood to be of sufficient scope to include those types of gears known as bevel gears, “hypoid” gears, as well as those gears known as “crown” or “face” gears. 
     A first embodiment of the inventive machine for manufacturing bevel gears is illustrated in FIGS. 1-5 and designated generally by  2 . For ease in viewing the various machine components, FIGS. 1-5 illustrate the inventive machine without doors and exterior sheet metal. The machine  2  comprises a single stationary column  4  of the type disclosed in U.S. Pat. No. 6,120,355, the disclosure of which is hereby incorporated by reference. Column  4  is preferably a monolithic structure such as cast iron but may be assembled from metal plates, for example steel plates, or may comprise individual frame elements such as corner posts and frame elements positioned as appropriate to support machine guideways or other components. Column  4  comprises at least two sides, preferably four sides, with at least two of the sides, first side  6  and second side  8 , being oriented at a desired angle, preferably perpendicular, to one another although sides oriented at angles greater than or less than 90 degrees (see column  4  in FIG. 19, for example) are also contemplated by the present invention. Each of the first and second sides comprises a width and a height (as viewed in FIG.  1 ). Alternatively, monolithic column  4  may comprise a form having non-planar sides such as, for example, a generally cylindrical column as illustrated by FIG.  17 . 
     First side  6  includes first spindle  10  having a front or seating surface  15 . Spindle  10  is rotatable about axis Q and is preferably driven by a direct drive motor  12 , preferably liquid-cooled, and preferably mounted behind front and rear spindle bearings (not shown). Spindle  10  is pivotably secured to a spindle support  11  which, along with spindle  10 , is movable in direction Z along the width of first side  6  on ways  14  attached to column  4 . Movement of spindle  10  in direction Z is provided by motor  16  through a direct-coupled ballscrew (not shown) or by direct drive. Preferably, a cutting or grinding tool  18  (cutting tool is shown) is releasably mounted to spindle  10  by suitable mounting equipment as is known in the art. 
     As stated above, first spindle  10  is attached to spindle support  11  such that pivoting of the spindle, and hence the tool  18 , may occur about pivot axis F. Spindle bracket  13  is pivotally attached to support  11  via at least one, and preferably two, bearing connections  20  and  22 , upper bearing connection  20  and lower bearing connection  22 . Pivoting of spindle  10  is effected by motor  24  and direct-coupled ballscrew  26 , or by direct drive, acting through sleeve portion  28  of yolk  30 . Yolk  30  is pivotally attached to spindle  10  preferably at an upper connection  32  and a lower connection  34  such that yolk  30  may angularly move relative to spindle  10  about axis V. Advancing of ballscrew  26 , and hence yolk  30 , effectively pushes drive motor  12  angularly away from column  4  thereby causing a pivot motion about axis F to angularly move the tool  18  toward the machine column  4 . See FIG. 3 for cutting a pinion and FIG. 5 for cutting a ring gear. Of course, retracting ballscrew  26  has the opposite effect. Alternatively, to effect pivoting of spindle  10 , a slide movable on at least one guideway oriented in the Z direction and positioned on spindle support  11  may be connected to spindle  10  or motor  12  via a linkage mechanism. Movement of the slide on the guideway effects pivoting of spindle  10  about axis F. A further alternative is to include a motor at one or both of bearing connections  22  and  23  to effect pivoting of spindle  10 . 
     Second side  8  includes second spindle  40  which is rotatable about axis N and is preferably driven by a direct drive motor  42 , preferably liquid-cooled, and preferably mounted behind front and rear spindle bearings (not shown). Spindle  40  is movable in direction X along the width of second side  8  on ways  44  attached to slide  46 . Movement of spindle  40  in direction X is provided by motor  48  through a direct-coupled ballscrew  49  or by direct drive. Preferably, a workpiece (a pinion  50  in FIG. 1 and a ring gear  51  in FIG. 4) is releasably mounted to spindle  40  by suitable workholding equipment  41  as is known in the art. Spindle  40  is also movable in direction Y along the height of second side  8  since slide  46  is movable in the Y direction via ways  52  with movement being provided by motor  54  through a direct-coupled ballscrew  55  or by direct drive. Directions X, Y and Z are preferably mutually perpendicular with respect to one another although one or more may be inclined with respect to its perpendicular orientation. For purposes of illustration, in all Figures, the Y direction is vertical. 
     While the arrangement of ways  44  and  52  is preferred as shown in FIGS. 1-5, FIG. 20 illustrates an alternative but less preferred embodiment where ways  44  may be attached to side  8  with slide  46  being movable in the X direction on ways  44 . Ways  52  may be arranged on slide  46  and spindle  40  attached to ways  52  for movement in the Y direction. It is also contemplated that movement in the Y direction may be effected by spindle  10  instead of spindle  40  (FIG. 18, with motors removed for clarity purposes). 
     The present invention makes possible, by use of a vertical column as the common support for both the workpiece and tool spindles, pivoting of the spindle on which the tool resides as shown in FIGS. 1-5. Conventional pivoting of the workpiece spindle is also technically possible, as is shown in FIG.  16 . Pivoting of the workpiece spindle, however, may require large pivot angles for ring gears resulting in a degradation of static and dynamic stiffness. With pinions, pivoting of the workpiece spindle is, at best, a compromise given that mounting distances, arbor heights and pitch angles vary over a wide range within pinions and even more when considering both pinions and ring gears. 
     Alternatively, both spindles  10 ,  40  may be pivoted as seen in FIG. 19 which shows a pivoting mechanism (e.g. yolk  30 ,  30 ′) attached for angular movement about axes (V, V′) to each spindle  10 ,  40 . While each spindle  10 ,  40  may actively pivot about respective pivot axes (F, F′) during manufacture of a gear, the present invention also contemplates one of the spindles  10 ,  40  being set at a predetermined pivot angle prior to manufacture of a gear, or, one of the spindles  10 ,  40  pivoting between incremental set positions during manufacture of a gear. Movement between such incremental set positions may reduce the amount or magnitude of pivoting necessary by the other of the spindles during manufacture of the gear. 
     Movement of first spindle  10  in direction Z, second spindle  40  in direction X, second spindle  40  via slide  46  in direction Y, pivoting of first spindle  10  about axis F, as well as first spindle  10  rotation and second spindle  40  rotation, is imparted by the separate drive motors  16 ,  48 ,  54 ,  24 ,  12  and  42  respectively. The above-named components are capable of independent movement with respect to one another or may move simultaneously with one another. Each of the respective motors is preferably associated a feedback device such as a linear or rotary encoder, such as pivot axis encoder  23  (FIG.  1 ), as part of a CNC system which governs the operation of the drive motors in accordance with instructions input to a computer controller (i.e. CNC) such as the Fanuc model 160i or Siemens model 840D (not shown). 
     The machine of the present invention as illustrated by the embodiments is guided by the controller which preferably continuously issues positioning and/or velocity commands to the various drive motors. Rather than load a large number of axis-positioning commands into the controller, it may be more efficient and meaningful to input a smaller set of data describing the gear manufacturing process. A logical candidate for such data is a set of “basic machine settings.” Using this approach, a machine operator would enter a set of basic machine settings (discussed in detail below) into the controller, which, in turn, would calculate the axis positions corresponding to a range of cradle positions. Thus, the basic “language” for describing bevel gear generating motions is retained in the present invention. 
     The relationship between the theoretical generating gear in mesh with a workpiece is maintained in the present invention by angular movement between the tool and workpiece axes in combination with relative rectilinear movements between the tool and workpiece axes along one or more of the three rectilinear axes and rotational movement of the workpiece about its axis. In the case of continuous indexing, rotational movement of the tool axis is also controlled. 
     Because of the complexity of tooth surfaces formed by conventional mechanical cradle-style bevel gear generators, such tooth surfaces can only be exactly defined geometrically by the machine motions which are used to produce them. While some general parameters of gear design may be specified (e.g. number of teeth, pitch angle, etc.) the equations which are used to define bevel tooth surfaces are the motion equations of generating machines. 
     Given the above, it is evident that with each machine configured differently than the conventional mechanical cradle-style bevel gear generator, a new set of formulas and other know-how would be required to determine appropriate machine settings and operating parameters for producing known gear tooth geometry and mating characteristics. However, since the conventional mechanical cradle-style bevel gear generating machine has been in existence for many years, a large amount of know-how already exists which relates desired tooth geometry and mating characteristics to conventional cradle-style machine settings. 
     Therefore, although a new set of formulas may be developed for a newly configured machine, it has generally become the practice in the art to utilize the same input parameters as a conventional mechanical cradle-style gear generating machine for other machines having a different number and/or configuration of axes. In other words, the positions of the tool and workpiece axes in the coordinate system of a conventional mechanical cradle-style bevel gear generating machine are transformed into the alternative coordinate system of the newly configured machine. An example of such a transformation can be found in U.S. Pat. No. 4,981,402 the disclosure of which is hereby incorporated by reference. The relationship between the invention and the conventional mechanical cradle-style bevel gear generator will be discussed below. 
     A conventional mechanical cradle-style bevel gear generating machine  60  (FIG. 6) for producing bevel gears generally comprises a machine frame  62 , work support mechanism  64  and a cradle support  66  comprising a cradle mechanism  68 . Traditionally, conventional mechanical cradle-style bevel gear generating machines are usually equipped with a series of linear and angular scales (i.e. settings) which assist the operator in accurately locating the various machine components in their proper positions. The following is a description of settings found on a tilt-equipped conventional mechanical cradle-style bevel gear generating machine such as the machine shown in FIG.  6 : 
     Eccentric Angle  70  controls the distance between the cradle axis, A C , and the tool axis, T, 
     Tool Spindle Rotation Angle  72  controls the angle between the cradle axis and the tool axis, commonly called the tilt angle, 
     Swivel Angle  74  controls the orientation of the tool axis relative to a fixed reference on the cradle  88 , 
     Cradle Angle  76  positions the tool  78  at some angular position about the cradle axis, 
     Root Angle  80  orients the work support  64  relative to the cradle axis, 
     Sliding Base  82  is a linear dimension which regulates the depth of tool engagement with the workpiece, 
     Head Setting  84  is a linear adjustment of the work support  64  along the workpiece axis, W, and, 
     Work Offset  86  controls the offset of the workpiece axis relative to the cradle axis. 
     A final setting, ratio-of-roll, governs the relative rotational motion between the cradle  68  and workpiece  88 . It should be noted that some of the above machine settings must be calculated taking into account the following workpiece and tooling design specifications: 
     the mounting distance of the blank workpiece (symbol−M d ), 
     the overall length of the work holding equipment (symbol−A b ), and, 
     the overall height of the tool (symbol−h). 
     Although the measures of these settings allow precise positioning of the machine components, the measures themselves convey little information about their location relative to one another. For instance, a head setting of 5 inches will position the work support in a different physical location relative to the cradle depending on the model of machine considered. This situation results from the “zero” head-setting position being defined differently on different model machines. In a similar manner, a setting of 30 degrees on the eccentric angle communicates little regarding the distance between the tool and the cradle axis since it is an angular measure which actually controls a linear dimension. Additional details must be furnished before the more meaningful linear distance can be calculated. 
     More immediately significant to the artisan is a set of absolute measures of machine component positioning, that is, measures which are independent of the tooling or machine model considered. These general, or basic machine settings, immediately communicate a sense of size and proportion regarding the generating gear and the workpiece being generated. They also provide a common starting point for gear design. For example, gear sets may be designed in terms of basic settings, thus unifying design procedures among many models of machines. In addition, analysis procedures need be written only once to cover all machine configurations if basic settings are employed. Of course, conversion to true machine-dependent settings is required to set-up a conventional mechanical cradle-style bevel gear generator but this is best performed just before presentation as a machine set-up summary. 
     A description of basic machine settings appears below and with reference to FIGS. 7-9. FIGS. 7 and 8 show, respectively, top and front views of a conventional mechanical cradle-style bevel gear generator with tilt. FIG. 9 is a projection showing a side view of the tool in true length. Details unrelated to the present discussion have been omitted for clarity. 
     Initially, two reference points are defined. The first point, point C T , is on the tool axis at some known position relative to the tool. This point, called the Tool Center, is usually chosen to lie in the plane defined by the tips of the tool (FIG.  9 ). The second reference point, CP, lies on the workpiece axis at the crossing point, that is, the point of intersection of the workpiece axis and the axis of its mating member. In the case of hypoid gears, CP lies at the point of apparent intersection between mating members when viewed in a plane parallel to both axes. Another point of interest, point O, is known as the machine center. This point is defined by the intersection of the cradle axis and the plane of cradle rotation (FIG.  7 ). 
     Using the above points, the following basic settings may be defined: 
     Radial, s, (FIG.  8 )—the distance from machine center O to tool center C T  when viewed in the plane of cradle rotation. 
     Cradle Angle, q, (FIG.  8 )—the angle formed by radial OC T  and a plane parallel to both the workpiece and cradle axes. 
     Tilt Angle, i, (FIG.  9 )—the angle formed by the tool axis and cradle axis. Usually taken to be between 0 and 90 degrees. 
     Swivel Angle, j, (FIG.  8 )—determines the direction of tool axis tilt. It is measured from line C T A which is rigidly connected and perpendicular to radial line OC T . Its measure is the angle formed by line C T A and the projection of the tool axis on the plane of cradle rotation. 
     Work Offset, E m , (FIG.  8 )—the minimum distance between the cradle axis and workpiece axis. 
     Sliding Base, X b , (FIG.  7 )—the distance between machine center O and point H, the point of apparent intersection of workpiece and cradle axes. This appears true length when viewed in a plane parallel to both cradle and workpiece axes. 
     Head setting, X p , (FIG.  7 )—the distance between apparent point H (identified above) and crossing point CP. Measured along the workpiece axis. 
     Root Angle, γ, (FIG.  7 )—the angle formed by the workpiece axis and the plane of cradle rotation. 
     Note: All parameters appear true length in the noted Figures, and positive in the sense shown. 
     The generation process is mainly governed by the ratio-of-roll (ratio of workpiece rotation to cradle rotation). Additional motion parameters (e.g. helical motion) may also be defined to augment the rolling motion between the cradle and workpiece. It is noted that other arrangements of basic machine settings could have been chosen instead of the one described. However, this particular choice of settings retains a likeness with conventional mechanical cradle-style bevel gear generating machine configurations, and clarifies essential geometric properties where appropriate. 
     Besides the eight settings defined above, it is useful to measure the rotational position of the workpiece about its own axis from some reference. Also, in the case of face hobbing, the rotary position of the tool about its own axis may be of interest. Combined together, these ten parameters totally describe the relative positioning between tool and workpiece at any instant. Three of them (cradle angle, workpiece rotation, tool rotation) change in the process of generation, while the other seven are “true” settings, i.e. they usually remain fixed. 
     A mathematical model is developed which accepts the basic machine settings, identified above, and exactly replicates bevel gear generation on the inventive embodiments through displacements along or about its six axes. FIG. 10 and 11 show, respectively, partial front and top views of the inventive tool and workpiece arrangement in the coordinate system of the first embodiment of the present invention. Referring to FIGS. 7-9, which illustrate the tool and workpiece arrangement of a conventional mechanical cradle-style bevel gear generating machine in the coordinate system of that conventional machine, vectors are defined along the workpiece and tool axes:                p   →     =     {         -   cos                   γ     ,   0   ,       -   sin                   γ       }             workpiece  axis                 c   →     =     {       sin                 i                   sin        (     q   -   j     )         ,                sin                 i                   cos        (     q   -   j     )         ,     cos                 i       }             tool  axis                         
     Next, the “key-way” vector, perpendicular and attached to the workpiece and tool axes, are defined:                a   →     =     {         -   sin                   γ     ,              0   ,     cos                 γ       }             workpiece  key-way  vector                 b   →     =     {       cos        (     q   -   j     )       ,     -     sin        (     q   -   j     )         ,   0     }             tool  key-way  vector                         
     Finally, a vector R is defined from the tool seat T R  (the back of the tool) to the point W R  on the workpiece axis which lies directly in the seating surface plane of the work arbor: 
     
       
           {right arrow over (R)}={−s  cos  q, s  sin  q−E   m   , X   b }−( X   p   +M   d   +A   b ) {right arrow over (p)}+h{right arrow over (c)}   
       
     
     Motions of the machine embodiment of FIGS. 1-5 may now be determined. A new coordinate system is associated with the axes arrangement of the orthogonal machine of FIGS. 1-5 with the origin being at point W R  on the seating surface or nose  43  of the machine spindle  40  Orthogonal axes are given by:                  u   →     X     =     p   →             workpiece  axis,  lines  up  with     X     axis                   u   →     Y     =     -         p   →     ×     c   →                p   →     ×     c   →                        vertical,  pointing  up,  lines  up  with     Y     axis                   u   →     Z     =         u   →     X     ×       u   →     Y                 horizontal  and  perpendicular  to                     u   x          ,  lines  up                             with     Z     axis                         
     Since pivot axis, F, as shown in FIGS. 1-5 is not located on the workpiece axis, as is customary, but instead is preferably positioned in the vicinity of the tool as shown by vector Δ 1  in FIG. 10, the position of the pivot axis in the new coordinate system must be defined. 
     With reference to FIGS. 12 and 13, pivot axis F is defined in a coordinate system attached to tool  18  in which the axis Z C  is coincident with the axis {right arrow over (c)} of the cutting tool and axis X C  is perpendicular to Z C  and extends along the back surface of the tool  18  (FIG.  12 ). The following can be seen from FIG.  12 :                  u   →     ZC     =     c   →               unit  vector                     u   ZC                     in  the  direction  of                     Z   C                     u   →     XC     =         u   →     y     ×       u   →     ZC                 unit  vector                     u   XC                     in  the  direction  of            X   C                     Δ   →     C     =     {       Δ                   x   C       ,   0   ,     Δ                   z   C         }                                       
     As seen in FIG. 13, transformation of {right arrow over (Δ)} C  in the tool coordinate system of FIG. 12 to the new coordinate system of the embodiment shown in FIGS. 1-5 is given by:                    Δ   →     1     =                      (     B   -     180   ∘       )     y     ·       Δ   →     C                     and       ,   therefore   ,                   Δ   →     1     =                  (                      cos        (     B   -     180   ∘       )           0         sin        (     B   -     180   ∘       )               0       1       0             -     sin        (     B   -     180   ∘       )             0         cos        (     B   -     180   ∘       )                        )          {           Δ                   x   C               0             Δ                   z   C             }                   =                {               -   Δ                     x   C        cos                 B     -     Δ                   z   C        sin                 B               0               Δ                   x   C        sin                 B     -     Δ                   z   C        cos                 B             }                           
     From the coordinate system of FIGS. 10 and 11, which represents the coordinate system of the embodiment illustrated in FIGS. 1-5, it may be seen that:            R   →     1     =         {             R   →     ·       u   →     X                   R   →     ·       u   →     Y                   R   →     ·       u   →     Z             }                   and                     R   →     2       =         Δ   →     1     -       R   →     1                         
     wherein: 
     R 1 =vector from point T R  on tool to point W R  on the seating surface  43  of machine spindle  40 , and, 
     R 2 =vector from point W R  on the seating surface  43  of machine spindle  40  to pivot axis F. 
     Therefore, the displacement along the X, Y, Z rectilinear axes of the machine embodiment of FIGS. 1-5 at a specified increment, such as each increment of generating roll, are calculated:                A   X     =     R     2   X               displacement  along     X     axis                 A   Y     =       R   →       2   Y               displacement  along     Y     axis                 A   Z     =       R   →       2   Z               displacement  along     Z     axis                         
     The three angular rotations must also be found. The pivot angle, B, at a specified increment, such as each increment of generating roll, is given by:        B   =     arccos        (       -     p   →       ·     (           p   →     ×     c   →                p   →     ×     c   →              ×     c   →       )       )                       
     The tool and workpiece axes each have an associated rotational phase angle which is superimposed on their motions as defined by conventional mechanical generators. These compensate for the changing relative orientation of conventional and inventive machine horizontal planes at a specified increment, such as each increment of generating roll. They are defined as:              α   =     arcsin        (       -     a   →       ·         p   →     ×     c   →                p   →     ×     c   →                )               workpiece  axis  phase  angle               β   =     arcsin        (       -     b   →       ·     (           p   →     ×     c   →                p   →     ×     c   →              ×     c   →       )       )               tool  axis  phase  angle                         
     An operation is also performed to determine the desired rotational position of the workpiece, ω, in accordance with phase angles alpha and beta and other setup constants including ratio of roll, R a , which specifies the ratio of relative rotation between the imaginary cradle and workpiece required for generation, indexing or hobbing constant, R C , which specifies the ratio of relative rotation between the tool and workpiece for continuous indexing, and reference constant ω 0  which specifies a known rotational position between the tool and workpiece. Other constants (not shown) may be used to further adjust the workpiece axis rotational position for duplicating special motions of conventional mechanical cradle-style machines such as “modified roll.” The operation may be expressed as: 
     
       
         ω=ω O   +f ( R   a   ,Δq )+ f ( R   C   ,Δt )+ f ( R   C , beta)+alpha  
       
     
     wherein: 
     Δq=q−q 0    
      with 
     q=instantaneous cradle roll orientation 
     q 0 =cradle orientation at center of roll 
     Δt=t−t 0    
      with 
     t=instantaneous tool spindle orientation 
     t 0 =initial tool spindle orientation 
     The above equation as written represents one embodiment of the general mathematical relationship wherein workpiece rotation is a function of R a , R C , alpha, beta, q and t. However, other variables such as intermediate variables in the form of basic settings s, i, j, E m , X b , X p , and γ, for example, may also be utilized in describing workpiece rotation resulting from input parameters. The calculation for ω is not limited to the specific expression shown above for this embodiment. 
     It has been discovered that the pivot axis F, defined, for instance with respect to cutting tools, within the cutting tool reference plane coordinate system X CR -Z CR  of FIG. 14, is preferably located in the quadrant of that coordinate system where X CR  is positive and Z CR  values are negative. Axis X CR  lies in the cutter reference plane  92  defined by the mid-point of the height of the blade cutting edges and axis Z CR  is coincident with the tool axis {right arrow over (c)}. Applying this definition to the embodiment of FIG. 1, for example, with axis Q perpendicular with axis N, it can be seen that the pivot axis F should be located on or “behind” the reference plane of the cutting tool  18  and at a point between the axis Q and the machine column  4 . Although the above positioning of the pivot axis is preferred, placement of the pivot axis along axis Q or outward from axis Q away from machine column  4  may be included in the present invention. 
     Placement of the pivot axis F should preferably be at a location whereby smooth and minimal motion along the axes is exhibited, such as noted on motion diagrams utilized to analyze machine motions, along with few, if any, reversal or inflection points. Preferably, pivot axis F should be positioned in the quadrant discussed above at a location therein defined by a positive ΔX CR  value being equal to the average radius of the cutting tool(s) to be used on the machine. Preferably, ΔZ CR  is equal to zero. For example, if cutting tools having diameters of 3 inches and 9 inches are contemplated, the average radius of the cutting tools would be 3 inches. Thus, ΔX CR  would be 3 inches, placing it at about point G in FIG. 14 if, for example, cutting tool  18  has a radius of 4.5 inches. Point G is in the vicinity of the gear tooth calculation point (for the average pinion or ring gear) which is located at the center of a tooth. A pivot axis passing through point G would be perpendicular to the X CR -Z CR  plane. 
     Also preferred is placement of the pivot axis in a location that allows the pivoting mechanism to be isolated from the workpiece and tool, such that it can be shielded from stray chips. Isolating the pivot axis should preferably still permit minimal and smooth motion along the axes with few, if any, reversal or inflection points as noted on machine motion diagrams as was discussed above. Given this, it has been found that one preferred location for the pivot axis F is at a point ΔX CR  located between the cutting blades of the largest tool contemplated for the machine and the machine column  4 , and at a ΔZ CR  generally about equal in magnitude to ΔX CR . More specifically, ΔX CR  is preferably at about the average diameter of the tools contemplated for the machine and ΔZ CR  is preferably generally about equal in magnitude to ΔX CR . For example, if cutting tools of 3 inch diameter and 9 inch diameter are contemplated, the average diameter is 6 inches. Thus, ΔX CR =6 inches, placing it beyond the cutting blades of the largest tool which would be at ΔX CR =4.5 inches for the 9 inch diameter tool. ΔZ CR  would also be generally about 6 inches but may vary plus/minus 2 inches. With placement of the pivot axis as set forth, travel of about 10-30 mm is noted along each of the linear axes which is desirably small and yet of a magnitude such that motion along the axes is accurately controllable by the machine controls. 
     As an example, a 12 tooth pinion having a pitch angle of 28.73° and a spiral angle of 50.0° is produced by generated face hobbing on a machine as shown in FIGS. 1-3. The basic settings for the machine were as follows: 
     
       
         
           
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 s = 135.82 
                 radial 
               
               
                   
                 q = 65.83 
                 center of roll 
               
               
                   
                 i = 31.79 
                 tilt angle 
               
               
                   
                 j = 320.26 
                 swivel angle 
               
               
                   
                 E m  = 48.2638 
                 offset 
               
               
                   
                 X p  = −0.0091 
                 head setting 
               
               
                   
                 X b  = 34.6578 
                 slide base offset 
               
               
                   
                 gamma (γ) = −0.01 
                 root angle 
               
               
                   
                 M d  = 116.84 
                 mounting distance 
               
               
                   
                 A b  = 139.7 
                 arbor height 
               
               
                   
                 h = 101.6 
                 tool height 
               
               
                   
                 BN = 17 
                 number of blade groups on the cutting tool 
               
               
                   
                 R a  = 3.58335 
                 ratio of roll 
               
               
                   
                   
               
            
           
         
       
     
     The hobbing or index constant, R C , is defined by the ratio of the number of blade groups on the cutting tool divided by the number of teeth on the workpiece. Therefore: 
     
       
           R   C   =BN /no.teeth workpiece =17/12  
       
     
     Additional machine constants (see FIG.  12 ): 
     ΔX C =152.4 mm 
     ΔZ C =−76.2 mm 
     Looking at the machine axes motion diagram of FIG. 15, it is shown that during the generation of the face hobbed pinion described above, there was about 20 mm of motion along each of the Z and Y axes and about 30 mm of motion along the X axis. It is also noted that rotation about the pivot axis F was about 0.5 degree. No points of inflection or reversal for any axes are noted on the diagram. 
     Conventionally, the workpiece is pivoted relative to the base. The introduction of the use of a single column to support both the tool spindle and the workpiece spindle now allows the tool spindle to be pivoted relative to the column. It may also be possible, however, for certain applications, to pivot the workpiece spindle either alone or in conjunction with pivoting the tool spindle. 
     It is to be understood that although the present invention has been discussed and illustrated with respect to a cutting machine, the present invention is also understood to equally encompass a grinding machine for bevel gears. 
     While the invention has been described with reference to preferred embodiments it is to be understood that the invention is not limited to the particulars thereof. The present invention is intended to include modifications which would be apparent to those skilled in the art to which the subject matter pertains.