ass. prof. Dr. Miloš Němček                                   x0       addendum modification coefficient of Fellow
                                                              x2       addendum modification coefficient of ring gear
                                                              z0       number of teeth of Fellow cutter
DOES A RESTRAINT GIVEN BY A NULL                              z2       number of teeth of ring gear
SIZE OF ROOT SPACEWIDTH OF AN                                  f2     pressure angle at the root cylinder
INTERNAL GEARING EXIST?                                       n       basic rack flank angle
                                                              t       transverse rack flank angle
                                                               w0     manufacturing pressure angle of Fellow cutter
                                                                       proportion (must be multiplied by a module)


                                                                        This task is easy when it is solving only for a
                                                              theoretical solitary ring gear. It is the same as a calculation
                                                              for an attainment of parameters for sharp teeth at external
                                                              gears. It means that always exist parameters for which we
                                                              can get a theoretical ring gear with a null root spacewidth.
                                                              But these parameters are valid only for this theoretical case.
                                                              In actual every ring gear meshes with pinion (external gear)
                                                              and it has to be manufactured. For next consideration we
                                                              will presume ring gear manufacturing by a Fellow cutter.
                                                              And in this case the situation with the root spacewidth of
                                                              real ring gear is quite different than for theoretical one.
                                                              There are two parts in this article. In the first part A solving
                                                              for theoretical ring gear will be showed. In the second part
                                                              B solving for a real ring gear will be showed.

                                                              A) SOLUTION FOR A THEORETICAL RING GEAR

                                                                        Both manufacturing and meshing with a pinion is
                                                              not considered. So it is not necessary to think of eliminating
                                                              backlash for meshing pair. For the root spacewidth it is then
                                                                                  4  x  tg                       
                                                              e f 2  d f 2             2     n 
                                                                                                      inv t  inv f 2  (1)
                                                                                     2  z2                           
                                                                                                                      

                                                              d f 2  d 2  2  mn  h * 2  x2
                                                                                       f                                 (2)
Keywords: internal gear, root spacewidth, addendum
          modification coefficient.
                                                                               d b2
                                                               f 2  arccos                                              (3)
USED SYMBOLS                                                                   df2

aw0    end working centre distance between Fellow cutter                The equation (1) has always one solution for
       and ring gear                                          ef2 = 0. This curve is similar to the curve of the tooth
da0    tip diameter of Fellow cutter                          thickness on the tip cylinder for outer gearing (this curve is
d2     pitch diameter of ring gear                            just the mirror image to the first one by y axe). The curve
db2    base diameter of ring gear                             for the dependence ef2 on the addendum modification
df2    root diameter of ring gear                             coefficient x2 is drawn in the fig. 1. This example is made
hf2    ring gear dedendum                                     for the ring gear with z2 = -80. The null root spacewidth is
ha0    Fellow cutter addendum                                 for addendum modification coefficient x2 = -2,24. The point
mn     normal module                                          for the root spacewidth ef2 = 0,2·mn is shown on this graph
sa0    tooth thickness on the tip cylinder of Fellow cutter   too. For the given ring gear this root spacewidth occurs for
                                                              addendum modification coefficient x2 = -1,66.
                                                                                                  Parameters of both cutters are designated by usual methods.
                                                            1             max.                    The standard basic rack profile will be used – whole cutter
                                                                    0,2                           depth ha 0  1,25 , teeth are straight.
-9                         -5                 -2                                     2      x2


                                                                                                  -9                   -5          -2                 2       x2

                                                                                                   Fig. 2 – ring gear manufactured by Fellow cutter with no
                                                                                                                 addendum modification coefficient
           Fig. 1 - theoretical ring gear with z2 = -80                                                                                     ef2

                                                                                                                                            e1f 2


         A solution is considered for a ring gear                                                 -9                   -5           -2                    2    x2
manufactured using a Fellow cutter. This ring gear meshes
with a pinion manufactured by any method. For calculation
of the root spacewidth are again used equations (1), (2) and                                      Fig. 3 – ring gear manufactured by Fellow cutter with sharp
(3). But there is one change. The root diameter df2 has to be                                                                   teeth
calculated from the end working centre distance of the                                            CONCLUSION
Fellow cutter.
                                                                                                           This article points to one important feature of an
                                                                                                  internal gearing manufactured by Fellow cutter. It is the
d f 2  2  a w0  d a 0                                                                   (4)
                                                                                                  basic difference between sizes of the root spacewidth of a
                                                                                                  theoretical ring gear and a real one manufactured by Fellow
Where basic Fellow cutter parameters are :                                                        cutter. The root spacewidth for a real ring gear is always
                                                                                                  greater then zero (in difference to the theoretical one). The
d a 0  mn  z 0  2  mn ha 0  x0                                                       (5)    minimum size of the root spacewidth can be the same as the
                                                                                                  tooth thickness on the tip cylinder of the used Fellow cutter
         mn  z 0  z 2  cos n
                                                                                                  - and moreover only with one exact addendum modification
a w0                                                                                     (6)    coefficient x2. With all other addendum modification
           2  cos         cos w0                                                               coefficients x2 is root spacewidth greater then the tooth
                                                                                                  thickness on the tip cylinder of the used Fellow cutter. This
             2  x0  x 2                                                                       feature of the real internal gearing significantly broadens a
inv w0                      tg n  inv n                                              (7)    range of available addendum modification coefficient x2,
                z0  z 2
                                                                                                  when an internal gearing is designed, in contrast to the
                                                                                                  theoretical internal gearing.
          Given to the fact then the Fellow cutter has always
on its tip cylinder the tooththickness greater then zero, it is
clear that the root spacewidth of the ring gear has to be
greater then zero too. In an extreme case, the root
spacewidth of the ring gear can be null one – for sharp teeth
                                                                                                  [1]      Němček,M.: Vybrané problémy geometrie čelních
of the Fellow cutter and moreover only for one addendum
                                                                                                           ozubených kol. MONTANEX a.s. Ostrava, 2003
modification coefficient x2. For given example (z2 = -80) are
                                                                                                           ISBN 80-7225-111-2.
designed graphs for their root spacewidth dependents on the
shape of the Fellow cutter. Two Fellow cutter will be used.
                                                                                                  [2]      Болотовский, И. А. и др.: Цилиндрические
Each has number of teeth z0 = 35, module will be mn = 1
                                                                                                           эвольвентные зубчатые передачи внутренного
                                                                                                           зацеплепия. МОСКВА, «Машиностроение».
a)   x 0  0; d a 0  37,5 [mm]; s a 0  0,49303                                 -       cutter
      without addendum modification coefficient (fig. 2)                                          AUTHOR

                                            *                                                     Miloš Němček, doctor of technical sciences, associate
b)   x 0  1,18072; d a 0  39,8614 [mm]; s a 0  0 - cutter
                                                                                                  professor – VŠB – Technical university of Ostrava, Czech
     with maximum allowable positive addendum                                                     republic
     modification coefficient - sharp teeth (fig. 3)

A generation of sharp teeth of external gears is a serious
limitation in their design. But an internal gear is a mirror
image of its external counterpart so one can assume the
same limitation with the narrow or null root spacewidth of
an internal gear. This article will provide an answer to this

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