Abstract:
A method of machining a bevel gear ( 22 ) having teeth with a pitch angle greater than 90 degrees comprising providing at least one tool ( 42, 44 ) rotatable about a tool axis with the tool comprising a disc-shaped body ( 43 ) having a periphery ( 49 ) and at least one stock removing surface ( 45, 47 ) arranged about the periphery. The tool is rotated and fed relative to the gear to effect machining of at least one tooth surface wherein the machining is carried out in a non-generating manner.

Description:
[0001]    This application claims the benefit of U.S. Provisional Patent Application No. 61/876,859 filed Sep. 12, 2013, the entire disclosure of which is hereby incorporated by reference. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The invention is directed to bevel gears and in particular to internal bevel gears. 
       BACKGROUND OF THE INVENTION 
       [0003]    High reduction transmissions can be realized with bevel ring gears that face each other with a shaft angle which is not equal to 180° (e.g. U.S. Pat. No. 7,147,583 to Lemanski). Such transmissions, known as pericyclic transmissions, comprise a system including a high reduction ratio, high tooth contact ratio, and nutating/rotating gear mechanism which incorporates meshing conjugate bevel ring gear or face-gear pairs. For example, the pericyclic transmission  2  shown in  FIG. 1  has an input shaft  4  which is connected to a bearing  6  inclined at a nutating angle. The outer ring of the bearing is connected to the pericyclic motion converter  8  which has teeth on both faces. The number of teeth on the pericyclic motion converter  8  differs by between 1 and 4 from the number of teeth of a reaction control member  10  which is rigidly connected to the housing  12 . Each rotation of the input shaft  4  will cause a nutation motion of the pericyclic motion converter  8  which is in mesh with the reaction control member  10  on the left side and an output gear  14  on the right side. The output gear  14  has the same number of teeth as the reaction control member  10 . 
         [0004]    If the number of teeth on the pericyclic motion converter  8  is, for example, 2 higher than the number of teeth of the reaction control member  10  (and the output gear  14 ), then each revolution of the input shaft  4  will rotate the output shaft  16  (which is rigidly connected to the output gear) by 2 pitches (i.e. 2 teeth). If the number of teeth on the output gear is, for example, 54, then the ratio of the transmission is i=360°/[(360°/54)*2]=27 (or 27×1). This shows that the principle of a nutating pericyclic motion converter can realize very high speed reductions in a very compact arrangement with a low number of rotating parts. The contact ratio between the teeth of the pericyclic motion converter  8  and the reaction control member  10  as well as the output gear  14  is very high. It is possible that 5 or more teeth participate on the motion transmission which enables the use of rather small gears for a high torque transmission. The low rotational motion of the pericyclic motion converter  8  will minimize the energy loss due to friction in the tooth meshing process and therefore results in a high transmission efficiency. 
         [0005]    It is possible to introduce additional rotation between the housing part  12 , which is rigidly connected to the reaction control member  10 , and the input shaft  4 . This motion can be actuated by an electric motor  18  with variable speed control which is controlled by an electronic circuitry. This addition effectively converts the high reduction transmission into an infinite variable transmission. 
         [0006]    Transmissions as described above have not been realized as production units in real industrial applications. Only prototypes have been manufactured using machining centers with ball nose end mills which required very long machining times and served the purpose of proving the concept of this transmission principle. The following observations have been made which explain the obstacle of industrial, cost-effective manufacturing of the gears especially the pericyclic motion converter  8 . 
         [0007]    The pitch cone apexes of the reaction control member  10  and the pericyclic motion converter  8  are coincident at the intersecting point of the axis of the two gear members which is given by physical law. An arrangement of this type is shown by the gears  20 ,  22  of  FIGS. 2( a ) and 2( b ) . This is also true if the number of teeth of the two facing ring gear combinations are different (as in the pericyclic transmission  2 ). If the two facing bevel ring gears have a shaft angle between 178° and 175°, then the pitch cone can be drawn as shown in  FIG. 2( b )  which means the reaction control member  10  (and the output gear  14 ) are external ring gears and the pericyclic motion converter  8  has internal bevel gears on both faces. For the purposes of discussion, external bevel gears are considered to be those gears having pitch angles of less than 90 degrees thereby having teeth that point outward and away from the axis of the gear. Internal bevel gears are considered to be those gears having pitch angles of greater than 90 degrees thereby having teeth that point inward toward the axis of the gear. 
         [0008]    Internal spiral bevel gears cannot be manufactured conventionally because the cutter has to approach the tooth from the back side ( FIGS. 2 c  and 2 d   ) of the gear and would generate the tooth instead the slot. The result would be complete destruction of the internal gear. Even if it was possible (according to bevel gear generating principles) to cut the internal gear from the front ( FIG. 2 c   ), the circular cutter path would cause secondary cuts and lead to mutilations which would destroy the teeth at the opposite side of the cutting zone (e.g.  FIG. 6 , interference area). 
         [0009]    Face gear solutions as indicated in  FIG. 3  where the external and the internal face gear member can be derived from the same shaper cutter have also been disclosed.  FIG. 3  shows a spur gear shaper cutter in a position where it generates an external face gear tooth on the left gear and in the same position an internal face gear tooth on the right gear. It can be observed that the shaper cutter approaches the external gear from the front (external), where the same section of the shaper cutter is supposed to form the teeth of the internal gear in the same position. This results in the shaper cutter destroying and eliminating the outer rim of the internal face gear in order to reach the tooth flanks from the back side. This relationship which is required in order to generate the correct mating flank to the external face gear cannot be realized in a practical manufacturing. 
         [0010]    The discussion above identifies some of the reasons that have prevented the conventional manufacturing of internal bevel gears. The existing prototypes have been manufactured either with three dimensional printing or with ball nose end mills on multi-axis machining centers. The machining center, for example, typically utilizes surface coordinates which can be calculated using a cutting simulation that works regardless of the fact that in actual practice, the parts would be mutilated or destroyed while cutting the slots from the back side. 
       SUMMARY OF THE INVENTION 
       [0011]    The present invention relates to a bevel gear set with a shaft angles below 180°, and preferably above 135°, which consists of an external and an internal bevel gear. The internal bevel gear is non-generated and is produced based on a complementary machine setup for generating the mating external gear. In order to manufacture the mating external gear member, the mirror image of the internal straight bevel gear is used as a generating gear. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0012]      FIG. 1  shows a three dimensional, partially exploded view of a high reduction pericyclic transmission. 
           [0013]      FIG. 2( a )  shows an external straight bevel gear to the left and an internal straight bevel gear to the right. 
           [0014]      FIG. 2( b )  shows the gears from  FIG. 2( a )  in a mesh position with coincident pitch cones. 
           [0015]      FIG. 2( c )  shows a two dimensional section of an internal ring gear having a generated profile with a tool presented from the front side and another tool presented from the back side. 
           [0016]      FIG. 2( d )  shows a two dimensional section of an internal ring gear having a non-generated profile with a tool presented from the front side and another tool presented from the back side. 
           [0017]      FIG. 3  shows a spur gear shaper cutter in a position where it generates an external face gear tooth on the left gear and in the same position an internal face gear tooth on the right gear. 
           [0018]      FIG. 4  shows a three dimensional view of a virtual plane generating gear and two interlocking cutters, forming one tooth of the generating gear. 
           [0019]      FIG. 5  shows a three dimensional view of the plane generating gear from  FIG. 4  in a horizontal arrangement with a straight bevel pinion generated on top and a straight bevel gear generated on the bottom side. 
           [0020]      FIG. 6  shows a three dimensional view of a schematic representation of an internal bevel gear and a circular face cutter tip track with the interference area. 
           [0021]      FIG. 7  shows a three dimensional view of a schematic representation of an internal bevel gear and two circular tip tracks of interlocking peripheral cutters. 
           [0022]      FIG. 8  shows the two dimensional top view onto the bevel gear basic machine (mathematical model) with a virtual conical generating gear and an internal straight bevel work gear. 
           [0023]      FIG. 9  shows the two dimensional top view onto the bevel gear basic machine with a virtual conical internal generating gear and an external straight bevel work gear. 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
       [0024]    The terms “invention,” “the invention,” and “the present invention” used in this specification are intended to refer broadly to all of the subject matter of this specification and any patent claims below. Statements containing these terms should not be understood to limit the subject matter described herein or to limit the meaning or scope of any patent claims below. Furthermore, this specification does not seek to describe or limit the subject matter covered by any claims in any particular part, paragraph, statement or drawing of the application. The subject matter should be understood by reference to the entire specification, all drawings and any claim below. The invention is capable of other constructions and of being practiced or being carried out in various ways. Also, it is understood that the phraseology and terminology used herein is for the purposes of description and should not be regarded as limiting. 
         [0025]    The details of the invention will now be discussed with reference to the accompanying drawings which illustrate the invention by way of example only. In the drawings, similar features or components will be referred to by like reference numbers. The use of “including”, “having” and “comprising” and variations thereof herein is meant to encompass the items listed thereafter and equivalents thereof as well as additional items. Although references may be made below to directions such as upper, lower, upward, downward, rearward, bottom, top, front, rear, etc., in describing the drawings, there references are made relative to the drawings (as normally viewed) for convenience. These directions are not intended to be taken literally or limit the present invention in any form. In addition, terms such as “first”, “second”, “third”, etc., are used to herein for purposes of description and are not intended to indicate or imply importance or significance. 
         [0026]    As discussed above,  FIG. 1  shows a transmission in a partially exploded view with an input shaft  4  which is connected to a bearing  6  that is inclined under a nutating angle. The outer ring of the bearing is connected to the pericyclic motion converter  8  which has teeth on both faces with the number of teeth differing by 1 to 4 from the number of teeth of the reaction control member  10  which is rigidly connected to the housing  12 . Each rotation of the input shaft  4  will cause a nutation motion of the pericyclic motion converter  8  which is in mesh with the reaction control member  10  on the left side and the output gear  14  on the right side. The output gear  14  has the same number of teeth than the reaction control member  10 . If the number of teeth of the pericyclic motion converter  8  is two (2) higher than the number of teeth of the reaction control member  10  (and the output gear  14 ), then each revolution of the input shaft  4  will rotate the output shaft  16  (which is rigidly connected to the output gear  14 ) by 2 pitches (i.e. 2 teeth). For example, If the number of teeth of the output gear is 40, then the ratio of the transmission is i=360°/[(360°/40)*2]=20 (or 20×1). This shows that the principle of a nutating pericyclic motion converter can realize very high speed reductions in a very compact arrangement, with a low number of rotating parts. The contact ratio between the teeth of the pericyclic motion converter  8  and the reaction control member  10  as well as the output gear  14  is very high. It is possible that 5 and more teeth participate on the motion transmission which enables the use of rather small gears for a high torque transmission. The low rotational motion of the pericyclic motion converter  8  will minimize the energy loss due to friction in the tooth meshing process and therefore result in a high transmission efficiency. 
         [0027]    As previously stated, it is possible to introduce an additional rotation between the housing part  12  (which is rigidly connected to the reaction control member  10 ) and the input shaft  4 . This motion can be actuated by an electric motor  18  such as an electric motor with variable speed control which is controlled by electronic circuitry. This addition effectively converts the high reduction transmission into an infinite variable transmission. 
         [0028]      FIG. 2( a )  shows an external straight bevel gear  20  to the left and an internal straight bevel gear  22  to the right. The difference in tooth count between the two gears is this example is one. In such as arrangement, the internal gear  22  has the larger number of teeth.  FIG. 2( b )  shows the gears from  FIG. 2( a )  in a mesh position with coincident pitch cones and the pitch apex coincident with the shaft intersecting point. The difference in the pitch cone angles is equal to the shaft angle Σ minus 90°. 
         [0029]      FIG. 2( c )  shows a two dimensional section of an internal ring gear  24  with a tool  26  presented from the front side and another tool  28  presented from the back side. The tools represents one tooth of a generating gear (generating rack) which move vertical in the drawing while the internal work gear rotates around its center. If the profile of the internal gear should be generated with the tool  26  presented from the front side, then the concave profile of the teeth  30  becomes mutilated entirely in the generating position. If the profile of the teeth  30  should be generated with the tool  28  presented from the back side, then the entire blank of the work gear  24  will be destroyed. 
         [0030]      FIG. 2( d )  shows a two dimensional section of an internal ring gear  32  having non-generated tooth profiles  34 . A tool  36  is presented from the front side and another tool  38  is presented from the back side. The tool  38  from the back side represents one tooth of a generating gear which is identical to the work gear (only plunging, no generating motion required). The tool  36  from the front side represents one tooth of a generating gear which is a mirror image of the work gear (only plunging, no generating motion required). If the profile should be formed with the tool  38  presented from the back side, then the entire blank of the work gear  32  will be destroyed. If the profile of the internal gear  32  should be formed with the tool  36  presented from the front side, then the correct flank form  34  is created without any risk of mutilation. 
         [0031]    The invention relates to a bevel gear set with a shaft angles below 180° and preferably above 135°, which consists of an external and an internal bevel gear. If a rotation is transmitted between two shafts with a low inclination angle (e.g.  FIG. 2 b   ), then the bevel gear with the larger number of teeth becomes an internal gear (γ 2 &gt;90°) in cases where the pitch angle of the gear with the lower number of teeth γ 1  is below the shaft angle Σ minus 90° (γ 1 &lt;Σ−90°). According to the gearing law as applied to bevel gears, the number of teeth of both members as well as the shaft angle have the following relationship: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Z 
                       1 
                     
                     
                       Z 
                       2 
                     
                   
                   = 
                   
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         γ 
                         1 
                       
                     
                     
                       sin 
                        
                       
                           
                       
                        
                       
                         γ 
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where:
       Z 1 =number of teeth of gear #1   Z 2 =number of teeth of gear #2   γ 1 =pitch angle of gear #1   γ 2 =pitch angle of gear #2       
 
         [0036]    The equation above presents two unknowns. In the case of a shaft angle, Σ, of 90 degrees, γ 1 +γ 2 =90° and 
         [0000]    
       
         
           
             
               
                 
                   
                     sin 
                      
                     
                         
                     
                      
                     
                       γ 
                       2 
                     
                   
                   = 
                   
                     
                       sin 
                        
                       
                         ( 
                         
                           
                             90 
                              
                             ° 
                           
                           - 
                           
                             γ 
                             1 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       cos 
                        
                       
                           
                       
                        
                       
                         γ 
                         
                           1 
                            
                           
                               
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     where 
                      
                     
                       : 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     
                       
                         Z 
                         1 
                       
                       
                         Z 
                         2 
                       
                     
                     = 
                     
                       
                         
                           sin 
                            
                           
                               
                           
                            
                           
                             γ 
                             1 
                           
                         
                         
                           cos 
                            
                           
                               
                           
                            
                           
                             γ 
                             1 
                           
                         
                       
                       = 
                       
                         tan 
                          
                         
                             
                         
                          
                         
                           γ 
                           1 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   and 
                    
                   
                     
 
                   
                    
                   
                     
                       γ 
                       1 
                     
                     = 
                     
                       arctan 
                        
                       
                         ( 
                         
                           
                             Z 
                             1 
                           
                           
                             Z 
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0037]    In the case of Σ≠90°, an iterative solution is utilized to determine the pitch cone angles as shown in the Table below: 
         [0000]    
       
         
               
             
               
               
             
               
             
               
               
             
               
             
               
               
             
               
             
               
               
               
             
               
             
               
               
             
               
             
               
               
               
               
             
               
             
               
               
               
               
             
               
             
               
               
               
               
             
               
               
               
             
               
               
               
             
               
               
               
               
             
               
               
               
             
               
               
               
               
             
               
             
               
               
             
               
             
               
               
               
               
             
               
             
               
               
             
               
             
           
               
                   
               
             
             
               
                 C —————————————————————————————————————————————————————————————   
               
               
                 C 
               
             
          
           
               
                   
                 SUBROUTINE PITCHCNE 
               
               
                 C 
                 =================== 
               
               
                 C 
                 ****************************************************************** 
               
               
                 C---- 
                 TEST PROGAMM FOR ITERATIVE PITCH CONE CALCULATION 
               
               
                 C---- 
                 IN CASE OF SIGMA =/= 90 deg 
               
               
                 C 
                 ****************************************************************** 
               
             
          
           
               
                 C 
               
             
          
           
               
                 C---- 
                 ITERATIVE SOLUTION OF THE FORMULA: 
               
             
          
           
               
                 C 
               
             
          
           
               
                 C 
                 SIN(GAMMA1)/SIN(GAMMA2) = Z1/Z2 
               
             
          
           
               
                 C 
               
             
          
           
               
                 C 
                 WHERE: 
                 GAMMA1+GAMMA2=SIGMA 
               
             
          
           
               
                 C 
               
             
          
           
               
                 C 
                 EXAMPLE: 
               
             
          
           
               
                 C 
               
             
          
           
               
                   
                 Z1 
                 = 54 
                  ! NUMBER OF TEETH REACTION CONTROL MEMBER 
               
               
                   
                 Z2 
                 = 55 
                  ! NUMBER OF TEETH GEAR 
               
               
                   
                 SIGMA 
                 = 177. 
                  ! SHAFT ANGLE 
               
             
          
           
               
                 C 
               
             
          
           
               
                   
                 GAMMA0 
                 = 90. 
                 ! ITERATION START VALUE 
               
             
          
           
               
                 C 
               
             
          
           
               
                   
                 DO 50 I 
                 = 1,9000 
                 ! ITERATION LOOP 
               
             
          
           
               
                   
                 GAMMA1 
                 = ASIN((Z1/Z2)*SIN(SIGMA−GAMMA0)) 
               
               
                   
                 GAMMA2 
                 = SIGMA−GAMMA1 
               
             
          
           
               
                   
                 IF ((ABS(GAMMA1−GAMMA0)).LT.0.00001) GOTO 55 
                 ! ABORTION CRITERIA 
               
             
          
           
               
                   
                 GAMMA0 
                 = GAMMA1 
                 ! RENEW n−1 VALUE 
               
             
          
           
               
                   
                 GAMMA2 
                 = SIGMA−GAMMA1 
               
             
          
           
               
                   
                 50 
                 CONTINUE 
                 ! NEXT STEP 
               
               
                   
                 55 
                 CONTINUE 
                 ! FINISHED 
               
             
          
           
               
                 C 
               
             
          
           
               
                 C---- 
                 RESULTS: 
               
             
          
           
               
                 C 
               
             
          
           
               
                 C 
                 GAMMA1 
                 =  69.195 deg 
                 ! PITCH ANGLE REACTION CONTROL MEMBER 
               
               
                 C 
                 GAMMA2 
                 = 107.805 deg 
                 ! PITCH ANGLE OUTPUT GEAR 
               
             
          
           
               
                 C 
               
             
          
           
               
                   
                 RETURN 
               
               
                   
                 END 
               
             
          
           
               
                 C —————————————————————————————————————————————————————————————   
               
               
                 C 
               
               
                   
               
             
          
         
       
     
         [0038]    The gear set described above can roll in mesh with conjugacy between their teeth which means the gear pair fulfills the gearing law and therefore rolls smooth with a constant ratio. The gear set is manufactured with the inclusion of lengthwise and profile crowning on the conjugate flank forms of both members in order to allow for load affected deflections. Internal bevel gearset have the potential for various industrial applications, such as pericyclic high reduction gearboxes. In the prior art, internal bevel gears have not been capable of manufacture according to the existing bevel gear manufacturing methods. The generating process of an internal bevel gear would require machining of the tooth slots not from the front, as schematically shown in  FIG. 2( c ) , left, but from the back side of the part (see  FIG. 2( c ) , right). Such a machining operation may be possible in theory to generate the flank surfaces, but in a real machining process it would destroy the back side of the gear blank and eventually eliminate the blank entirely. 
         [0039]    The inventors recognized that in case of a non-generated straight bevel gears, the reference profile is a trapezoidal channel as can be seen in  FIG. 4  which shows a three dimensional view of a virtual plane generating gear  40  and two interlocking cutters  42 ,  44  forming one tooth of the generating gear. Each cutter comprises a cutter body  43  having a plurality of cutting blades  45  arranged about the periphery  49  of the cutter body. Each cutting blade  45  has a cutting edge or surface  47 . The trapezoidal channel, which is formed by the two cutters, is repeatedly formed in rotational distances of one pitch around the circumference of the plane generating gear  40 . It was discovered that the trapezoidal channel has the same identical form on the front (e.g. Side 1) as it has on the back (e.g. Side 2). See  FIG. 4 , trapezoidal channel before and after φ=180° rotation. In cases of radially proportional channels, it is possible to form the channels not only from side 1 but also from side 2. A rotation of 180° around the X 4  axis will place the channel to the indicated position on side 2. It was further discovered that limited to non-generated bevel gears, any tooth geometry on a plane or conical bevel gear can be theoretically cut from both sides, realizing the identical flank form by utilizing the original tools, defined for the cutting from the back side. 
         [0040]      FIG. 5  shows a three dimensional view of the plane generating gear  40  from  FIG. 4  in a horizontal arrangement with a straight bevel pinion  46  generated on top and a straight bevel gear  48  generated on the bottom side. The tooth profiles of the generating gear  40  are trapezoidal channels that change their width and height from the outside to the inside proportional to the radial distance to the rotation axis of the generating gear. If the virtual generating gear is visualized to consist of a very thin material, then the tooth thickness and the slot width are equal on both sides. This makes it possible to generate a first gear on the top side and a second gear (with equal or different number of teeth on the bottom side of the same generating gear. As a consequence of the kinematic coupling conditions, established with this arrangement, the gear generated on the top side will mesh conjugate with the gear generated on the bottom side. 
         [0041]    Since cutting of an internal bevel gear from the (originally correct) back side is physically not possible, the inventive solution is the cutting of internal straight bevel gears from the front side, which is the side where the teeth are exposed (see  FIG. 2( a )  for example) using identical tool geometry and a modified machine setup. Another element required to manufacture internal bevel gears is the avoidance of interference between the tool and the work piece in areas away from the cutting zone. This interference is not avoidable in case of face cutting (or grinding) tools. The tool tip track will cause destruction of a workpiece.  FIG. 6  shows a three dimensional view of a schematic representation of an internal bevel gear and a circular face cutter tip track. Although this face cutter approaches the internal gear from the front side, it will create an interference area about opposite to the cutting zone which is caused by the angular inclination to fit the circular path of the face cutter to the internal zone and the fact that the blades which exit the cutting zone have to rotate along a circular track. 
         [0042]    Straight bevel gears produced with interlocking cutters of the type shown in  FIG. 4  use peripheral cutters where the flank forming side of cutter and blades envelopes a plane which is perpendicular to the cutter axis. The two planes of the two cutters that form both flanks represent the surfaces of a trapezoidal channel as one tooth of the generating gear (see  FIG. 4 ).  FIG. 7  shows a three dimensional view of a schematic representation of an internal bevel gear and two circular tip tracks of interlocking peripheral cutters. The cutting edges of the peripheral cutters envelop a plane (the upper cutter on the top side and the lower cutter on the bottom side) which represent the “side walls” of the trapezoidal channel shown in  FIG. 4 . Interferences between the two cutters and the gear blank can be avoided since the cutter inclination is less than 45° with respect to the horizontal Y 4 -Z 4  plane, and the cutter tip tracks move away to the front of the internal ring gear. 
         [0043]    In case of cutting an internal gear as in  FIG. 7 , the cutters represented will not cause any interference. If the connected upper and lower cutters would perform a generating roll motion around axis Y 4  in  FIG. 7  then this would destroy the internal ring gear from the inside. Only the non-generated cutting tool setup, with peripheral cutters for straight bevel gears with the cutter positioned at the inside of the gear, adjacent to the slots will avoid mutilation and blank destruction. 
         [0044]      FIG. 8  shows the two dimensional top view onto the bevel gear basic machine (mathematical model) with a virtual conical generating gear and an internal straight bevel work gear, formed by the generating gear, without generating motion because the machine root angle Σ is 180° and the number of teeth of the generating gear and the work gear are identical. In case of a generating gear rotation, the work gear rotates with it without any relative motion.  FIG. 8  shows an arrangement of a basic bevel gear machine with the vector EX which positions the center of the peripheral cutter and the cutter radius vector RW, and which points below the root at the center of the face width. EX is arranged such that the arc of RW intersects the theoretical root line at the heel and toe boundaries. The machine root angle Σ, between work gear axis Z 1  and the generating gear axis Y 4 , is 180°. 
         [0045]      FIG. 9  shows the two dimensional top view onto the bevel gear basic machine with a virtual conical internal generating gear and an external straight bevel work gear. The teeth of the work gear are formed by the generating gear with a generating motion, because the machine root angle Σ is 177° (as shown by the example in the above Table) and the number of teeth of the generating gear and the work gear are different ( 55  and  54  respectively). In case of a generating gear rotation, the work gear rotates around a different axis than the generating gear which causes a relative (generating) motion. 
         [0046]    The requirement for the non-generated teeth created according to  FIG. 8  to roll correct and conjugate with the mating gear is a complementary setup of the machine which generates the mating gear. In order to manufacture a mating external gear member, the mirror image of the internal straight bevel gear is used as generating gear. This can be realized with the arrangement in  FIG. 9 . The cutter center is positioned at the tip EX. The cutter radius vector points at the center of the face width and the tip track intersects with the theoretical root line and the toe and heel boundaries. The machine root angle Σ is identical to the root angle between the internal and the external gear in their gearbox. According to the common rules of the gearing law, this root angle, in combination with the internal generating gear, requires a certain number of teeth. If the calculated number of teeth is not an integer, then an adjustment of the root angle is required to achieve an integer number of teeth. The rotation of the generating gear around axis Y 4  and a rotation of the work gear around Z 1  will cause a relative motion (generating motion) which will form external gear teeth which are exactly conjugate to the non-generated teeth of the internal member ( FIG. 8 ). The generating gear in  FIG. 9  represents the internal gear member, which is the main condition for the fulfillment of the gearing law between the two members and the conjugate rolling properties. 
         [0047]    The internal bevel gear of the invention may be produced on any applicable gear manufacturing machine, preferably the multi-axis CNC gear manufacturing machine disclosed in U.S. Pat. No. 6,712,566 the disclosure of which is hereby incorporated by reference. A preferred cutting arrangement is to form each tooth slot with a single rotary disc-shaped cutter (e.g.  42  or  44  of  FIG. 4 ) utilizing a first machining position for forming a first portion of the tooth slot and then repositioning the cutter to a second machining position for forming the remaining portion of the tooth slot. Such a method is disclosed in U.S. Pat. No. 7,364,391 the disclosure of which is hereby incorporated by reference. 
         [0048]    A preferred cutter for producing internal bevel gears is a rotary disc-shaped solid cutter having straight cutting edges (surfaces) about the periphery thereof (e.g.  42  or  44  of  FIG. 4 ) wherein the cutting edges envelope a plane which is perpendicular to the cutter axis thereby forming the straight “side walls” of the trapezoidal channel (e.g. as seen in  FIG. 4 ) during the non-generating cutting process. However, the cutter may also be a peripheral cutter utilizing replaceable inserts or stick-type cutting blades with straight cutting edges thereby also enveloping a plane. Alternatively, the cutting edges may be curved resulting in profile curvature of the tooth surface. Additionally, the cutter axis may be tilted, thereby enveloping an inner cone, to effect lengthwise crowning (curvature) of the tooth surface. 
         [0049]    The invention also contemplates finishing (e.g. grinding) of internal bevel gears with a peripheral tool wherein, for example, an abrasive material is arranged about the periphery of a disc-shaped tool body. Alternatively, the cutting blades discussed above may be replaced with finishing blades (e.g. skiving blades). 
         [0050]    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 without deviating from the spirit and scope of the appended claims.