METHOD FOR CUTTING STRAIGHT BEVEL GEARS
USING QUASI-COMPLEMENTARY CROWN GEARS


Koreaki Ichino
Nagaoka Gear Works Co., Ltd
Nagaoka, Japan

Hisashi Tamura and Kazumasa Kawasaki
Department of Mechanical and Production Engineering
Niigata University
Niigata, Japan


  ABSTRACT

    In the manufacture of straight bevel gears, there is usually a need for some 
  trial-and-error process by which a good tooth bearing is obtained.  In this 
  paper, a new straight bevel gear cutting method in which trial-and-error process 
  is not necessary is proposed.  The method can be applied at once without the 
  necessity of generator alteration.   Furthermore, each gear and pinion is 
  generated by a newly introducedgquasi-complementary crown gearh.  Gears made
  on an experimental basis showed the expected good tooth bearing.


  1. INTRODUCTION

    The tooth surface of a straight bevel gear is a simple surface as compared
  with that of a spiral bevel gear; nevertheless, trial-and-error process is still 
  employed to obtain a good tooth bearing in the straight bevel gear manufacturing.
  The authors think the trial-and-error process actually covers the defect of gear
  cutting method.  The gear cutting method based on the kinematic theory employs 
  the pitch cone of the generated gear and the pitch plane of the complementary 
  crown gear which are arranged to roll with each other.
    The straight bevel gear cutting method has been developed by Gleason Works
  [1, 2].  However, the method using the generator such as the Gleason No. 14 
  straight bevel gear generator, employs the members which are not arranged to 
  roll.  As a result, the real tooth surfaces generated in machining are slightly 
  different from the theoretical surfaces and the tooth bearing is unsuitable for
  practical use when those surfaces are meshed.  The unsuitable tooth bearing can 
  be corrected by the trial-and-error process of the skilled worker. 
    In this paper, a new method for cutting straight bevel gears is proposed.  
  The method employs the pitch cone of the generated gear and the pitch cone of a
  newly introducedgquasi-complementary crown gearhwhich are arranged to roll 
  with each other and keeps the kinematical strictness of gear tooth generation. 
  The pitch cone angle of the quasi-complementary crown gear is designated strictly 
  as 90 minus the root angle of the gear to be cut.  The tool surface of the 
  quasi-complementary crown gear is the plane with pressure angle .  The plane
  tool surface produces the modified tooth surface with depthwise tooth taper.  
  Besides, angle  determines the position of tooth bearing on the generated gear
  tooth surface.
    In the previous papers [3, 4], a method using a quasi- complementary crown
  gear in spiral bevel gear cutting was proposed.  The quasi-complementary crown
  gear can be realized by the Gleason No. 116 spiral bevel gear generator and can 
  generate only the modified tooth surface with depthwise tooth taper.  However, 
  the position of tooth bearing was controlled by another technique.
    The proposed method in this paper can be used in combination with the usual 
  crowning process in pinion machining.  The process of this method is the same as 
  the usual process except that the ratio of roll and the pressure angle of the 
  tool are theoretically determined taking the gear generation method and the 
  position of tooth bearing into consideration respectively, as stated above.  
  This method can be applied at once without the necessity of any alteration of 
  the usual generator.


  2. QUASI-COMPLEMENTARY CROWN GEAR

    In general, the complementary crown gear which generates the straight bevel
  gear has a depthwise tooth taper.  Therefore, it is necessary for the tool which
  simulates the crown gear tooth to move tooth-depthwise, namely, perpendicularly 
  to the cradle plane [5] theoretically.  This tool movement is in need of the 
  mechanism of tool inclination to cope with the various root angles of the gear 
  to be cut and complicates the machine, so that the workpiece is set to the

                           
                        Fig. 1  Relations between three gears


                           
                        Fig. 2  Clearance between tooth surfaces


  machine with more inclination angle than its pitch cone angle in practice to get
  around the machine complexity.  More inclination angle of the workpiece is the 
  root angle f of the gear to be generated.  However, this additional inclination
  of workpiece breaks the kinematical strictness of gear tooth generation.  
  In other words, the arrangement that the pitch cone of the workpiece and the 
  pitch plane of the complementary crown gear roll on each other is broken by this
  workpiece inclination.  The broken arrangement is strictly restored by the 
  quasi-complementary crown gear as follows:
    First, we call the complementary crown gear the crown gear for short.  Next,
  we imagine that the crown gear with the plane tooth surface xc generates both 
  the workpiece and gear-I with a large pitch cone angle at the same time as shown
  in Fig. 1.  The pitch plane of the crown gear lies between the workpiece and 
  gear-I, keeps in contact with each pitch cone, and rolls on each cone [6].  
  The pitch cone angle of the workpiece is 0 and the large pitch cone angle of 
  gear-I is ( 90-f ).  The ratio of roll between the crown gear and workpiece 
  is 1/sin0 from the condition of roll.  The ratio of roll between the crown gear
  and gear-I is 1/cosf .  Therefore, the ratio of roll i between the gear-I and
  workpiece is:

                                  i = cosf / sin0                          (1)

    These three gears are conjugate gears, so that gear-I can generate the same 
  gear with pitch cone angle 0 that the crown gear generates.  The gear-I is a 
  straight bevel gear with the tooth surface 

                            
                Fig. 3  Quasi-complementary crown gear and workpiece


  xT of octoid profile.  The tooth surface xT is very close to a plane because
  the pitch cone angle of gear-I is close to 90.  Figure 2 shows the surface xT 
  is in contact with the plane tooth surface xc of the crown gear at point Q.  
  There is a gap between xT and xc with the exception of point Q.  The gap is very
  small in the neighborhood of point Q.  Therefore, the real plane tool surface x 
  substitutes for xT as the tooth surface of gear-I.  A part of the substituted 
  tool plane x in the neighborhood of point Q can generate a conjugate tooth 
  surface in the workpiece.  The gap between the tooth surfaces away from point Q 
  is not so large that a part of the tool plane x away from point Q can generate a 
  modified tooth surface.  In this study, the gear-I with tool plane x is called 
  the quasi-complementary crown gear, namely, the quasi-crown gear for short, which
  generates the tooth surface of the straight bevel gear with modified profile and 
  depthwise tooth taper.
    The characteristics of the quasi-crown gear is as follows:
     1. The pitch cone angle is ( 90-f ).
     2. The ratio of roll is cosf / sin0.
     3. The tool surface is a plane as usual.
     4. The generated tooth profile becomes the modified one.
     5. The pressure angle of the quasi-crown gear is theoretically determined 
        relating to the position of tooth bearing as state in next section.
    Figure 3 shows the relation between the quasi-crown gear and workpiece.  In the
  machining of workpiece, the cradle axis is the quasi-crown gear axis, the set 
  angle of the workpiece axis is angle (0 - f ), and the straight sided tool is
  employed as usual.  However, the setting of the ratio of roll i is only different
  from the usual setting.
    From above, it is clear that the quasi-crown gear is realized easily by the 
  employment of an ordinary gear generator such as the Gleason No. 14 straight 
  bevel gear generator which is the most fundamental machine, without any 
  alteration.

                           
                  Fig. 4  Engagement of teeth in transverse plane

                           
              Fig. 5  Pressure angle    of quasi-complementary crown gear


  3. TOOTH BEARING AND PRESSURE ANGLE

    In the proposed method, a pair of quasi-crown gears is employed instead of a 
  crown gear except the case of miter gear generation.  One is for gear generation
  and the other is for pinion generation.  However, the crown gear is a fundamental
  gear in this method and is very useful for the analysis of the relationship
  between the position of tooth bearing and the pressure angle of each quasi-crown 
  gear.
    Figure 4 shows the crown gear tooth, the pinion tooth, and the gear tooth each 
  engage at point Q on the common transverse plane of the bevel gear.  The
  kinematical meaning of  point Q is the same as that of Fig. 2.  That is, the
  parts of both the pinion tooth surface and gear tooth surface which are away from
  point Q are the modified non-conjugate tooth surfaces causing a small 
  transmission error, and the parts of both the tooth surfaces in the neighborhood
  of point Q are the conjugate surfaces, so that the tooth bearing always appears 
  about point Q when a pair of generated gears are meshed.  Thus, the position of 
  point Q on the crown gear tooth surface represents the position of tooth bearing 
  on the meshed tooth surfaces, and besides, relates the tool pressure angle  of 
  each quasi-crown gear as shown  

                          
                      Fig. 6  Coordinate systems of each gear


  in Fig. 5.
    The tooth surface xc of the crown gear which rotates about its axis by angle 
   is used as the plane tool surface x of the quasi-crown gear which rotates 
  about its axis by angle .  The pressure angle  of tool surface x is unknown.
   is determined as follows:
    Referring to Fig. 4, we can introduce the following equation according to the 
  condition of meshing of spur gear kinematics:

                               H = - L sinccosc                            (2)

  where L is the moving length of tooth surface xc in the transverse plane, c is 
  the pressure angle of the crown gear and H (positive) is the height of point Q 
  from pitch plane S toward addendum.  H (negative) is toward dedendum.
    The length L can be expressed using the mean cone distance Rm and angle :

                                    L = Rm                                   (3)

  From Eqs. (2) and (3), the rotation angle  of the crown gear is:

                          = - H / (Rm sinccosc )                         (4)

    Three coordinate systems shown in Fig. 6 are introduced to analyze the 
  relationship between xc and x.  O-XYZ is fixed in space, O-xcyczc is attached to
  the crown gear and O-xyz is attached to the quasi-crown gear.  O-xcyczc and O-xyz
  rotate by angles  and  about axis zc and axis z respectively.  When both  
  and  are zero, axes xc and x agree with axis X, and yc agrees with axis Y which
  is the pitch surface generator [6].  In Y-Z plane, axis z is inclined by angle 
  f which is the root angle of the gear to be cut.
    In each coordinate system, nc represents the unit surface normal of xc which is
  the simulated plane of the crown gear tooth surface as shown in Fig. 7.  nc is 
  changed in O-XYZ only by angle  expressed as a function of H, Rm, and c.  n 
  represents the unit surface normal of x which is the real tool plane of the quasi
  -crown gear as shown in Fig. 8.  In O-XYZ, the surface xc attached to O-xcyczc 
  coincides with 

                         
                 Fig. 7  Tooth surface of complementary crown gear

                          
                Fig. 8  Tooth surface of quasi-complementary crown gear


  the surface x attached to O-xyz, namely, nc coincides with n.  Therefore, the 
  following equation is obtained:

                  A(-f ) C() n() = C() nc                              (5)

  where A and C are the coordinate transformation matrices and are as follows:

                                1          0         0   
                  A(f) = m   0      cosf     -sinf n
                                0      sinf      cosf
                                                                               (6)
                               cos     -sin      0
                  C()  = m  sin      cos      0    n
                                0          0         1

  Two unknowns  and  are determined as the solutions of Eq. (5).  The solutions
  are as follows:

                  = sin-1 (sincosc sinf + sinc cosf)
                                sincosc cosf - sinc sinf             (7)
                  = tan-1m \\\\\\\\\\\\\\\\\\\ n                            
                                         coscosc     

  According to Eqs. (4) and (7), the pressure angle  of the quasi-crown gear is 
  determined directly by H which is the position of tooth bearing.
    The pressure angle ' of mating quasi-crown gear is as follows:

                ' = sin-1(-sincosc sinf' + sinc cosf')             (8)

                   Table 1  Dimensions of straight bevel gears
                        
 
                Table 2  Dimensions of quasi-complementary crown gears

                         

  where f' is the root angle of mating gear.
    Now, the crown gear is not actually employed in this method, so that the 
  pressure angle c of the crown gear can be omitted from Eqs. (7) and (8), 
  replacing sin with .  As a result, the following equation is introduced:

                          a2 + b2 + c = 0                                  (9)

  where  = sin', a = - sinf cosf , b = sin(f -f')sin,    
                                                                              (10)
         c = sinf 'cosf' sin2 + H sin2(f +f') / Rm
                            
    The pressure angle ' is determined from Eq. (9) when f, f', H, Rm and an
  arbitrary angle  are given.  That is, the position of tooth bearing is 
  determined by the designation of ' corresponding to .
    In case of miter gear, for example, f = f' and H = 0 yield, so that the 
  result of ' =  is obtained.  This means that, according to this method, the 
  tooth bearing appears on the pitch cone (H = 0) without fail when a pair of gears
  is generated using the same straight sided tool with an arbitrary pressure angle.



  4. AMOUNT OF PROFILE MODIFICATION

    As stated in section 2, there is a gap between the theoretical tooth

                          
                           Fig. 9  Backlash and kickout


  surface of gear-I with octoid profile and the tooth surface of the quasi-crown 
  gear (see Fig. 2).  The amount of gap becomes that of the profile modification 
  because a small gap directly turns to the generated tooth surface.  Furthermore,
  the amounts become essentially the definite amounts according to the root angle
  f of the gear to be cut.  The amount of gap at arbitrary point on x is obtained
  from the following vector equation:

          x(u, v) + t n() = xT(U, )                                      (11)

  where U, , and u, v are the parameters to express the surface of xT and x 
  respectively, t is the amount of gap or the amount of profile modification, n is 
  the unit surface normal of plane x, and  is the known constant as stated above.
  The formulation and analysis of the generated tooth surface such as worm gear 
  surface, hypoid gear surface, spiral bevel gear surface, and straight bevel gear 
  surface xT in this study, have been developed by many researchers [7-18] using 
  the tool geometry, machine tool kinematics, differential geometry, theory of 
  conjugate surface and technique of coordinate transformation matrix.


  5. EXAMPLE OF MANUFACTURING STRAIGHT BEVEL GEARS
 
    The dimensions of the gears manufactured as an example are shown in Table 1.  
  These dimensions are designated according to Gleason method.  The root angle of 
  the gear is very large angle 851' and dedendum is twice the length of addendum. 
  If the position of tooth bearing is designated as a point 1 mm apart from the 
  pitch cone toward dedendum, the tooth bearing will appear at the middle of the 
  working depth of the gears.  Therefore, the gear manufacturing in two cases is 
  carried out.  One is the tooth bearing on the pitch cone (H = 0 mm) and the other
  is H = -1 mm on the gear tooth (H = +1 mm on the pinion tooth).  The 
  corresponding quasi-crown gear dimensions are shown in Table 2.
    The practical inspections of bevel gears are the check of the position of tooth
  bearing by the eye-measurement and the check of commonly calledgkickouthwhich 
  is a difference between the backlashes measured in two cases shown in Fig. 9.  
  It is desirable that the amount of kickout is zero.
    In case of H = 0 mm, the kickout is zero and tooth bearing appears


                           
                 Fig. 10  Amounts of profile modification [m ]
                              (without crowning)
  

  apart from the middle of tooth depth as arranged previously.  In caseof H = -1 mm
  on the gear tooth (H = +1 mm on the pinion tooth), the kickout is undesirably 
  large value 0.07 mm, but the position of tooth bearing is the middle of tooth 
  depth as arranged.
    The amount of kickout of the gears generated by this method can be calculated 
  from the amount of profile modification.  The amount of modification t is 
  calculated in case of the manufactured gears and the results are shown in Fig.10.
  The points A, A', B, and B' in Fig. 10 nearly correspond to the points in Fig. 9 
  respectively.  The profile modification causes the additional backlash and two 
  times the amount of modification becomes the amount of kickout.  Therefore, in 
  case of H = 0 mm, (AC + B E = ) 10m of additional backlash minus (A'C + B'E
  = ) 10m is zero kickout which coincides with experimental result.  In case of
  H = -1 mm on gear tooth and H = +1 mm on pinion, (A 14 + B 24 = ) 38m minus (A'
  0 + B' 0 = ) 0m is 38m which is nearly one half of 0.07 mm of experimental 
  result.  There are good agreements in two cases.
    In case of H = -1 mm of both gear and pinion, (A 0 + B 24 = ) 24m minus 
  (A' 14 + B' 0 = ) 14m is 10m, so that twice 10m will be the kickout.  
  A pair of gears with H = -1 mm was  manufactured, and the position of tooth 
  bearing and the kickout were

                            
                     Fig. 11  Tooth bearing on gear tooth surface
                           (Gear H = -1 mm, Pinion H = -1 mm)


  checked.  Figure 11 shows the tooth bearing which is very close to the previous 
  arrangement.  The measured kickout is about 0.015 mm which is slightly smaller 
  than the simulation result.


  6. CONCLUSION

    In this paper, a new method for cutting straight bevel gears was proposed.  
  In this method, the gear and pinion are generated by a correspondingly 
  quasi-complementary crown gear which is a newly introduced tool gear instead of 
  a usual complementary crown gear.  The proposed method is a simple one and can be
  applied at once without any alteration of a usual gear generator or the necessity
  of trial-and-error process in manufacturing.  This method is based on the 
  kinematics of gear generation.  Therefore, the ratio of roll, the pressure angle
  of the tool surface which is a quasi-complementary crown gear tooth surface, the 
  machine setting of workpiece, and the amount of tooth profile modification can be
  theoretically determined.  The usefulness of the proposed method was confirmed by
  some experiments of gear manufacturing.





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