ass. prof. Dr. Miloš Němček x0 addendum modification coefficient of Fellow cutter 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) INTRODUCTION 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 valid: 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. ef2 Parameters of both cutters are designated by usual methods. 1 max. The standard basic rack profile will be used – whole cutter -1,66 * -2,24 0 0,2 depth ha 0 1,25 , teeth are straight. -9 -5 -2 2 x2 ef2 1 min. 0 -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 1 B) SOLUTION FOR A REAL RING GEAR min. 0 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 LITERATURE greater then zero too. In an extreme case, the root spacewidth of the ring gear can be null one – for sharp teeth  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.  Болотовский, И. А. и др.: Цилиндрические Each has number of teeth z0 = 35, module will be mn = 1 эвольвентные зубчатые передачи внутренного [mm] зацеплепия. МОСКВА, «Машиностроение». 1977. * 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) ANNOTATION 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 question.