The questions of tooling design for production of advanced gears are considered. Engineering is based on the special applied development of the mathematical theory of multiparametric mappings of space. In fulfilled engineering of gear cutting tools for shaping of noninvolute gears it is provided for exclusion of distorted profiling after tool regrinds. There are proposed calculation algorithms, which may be used in dataware of respective CAD/CAM systems of maintenance for tooling backup. Among developed tools there are assembled shaping cutters with prismatic and round cutters. Compensatory possibilities of proposed assembled shaping cutters are ensured by repositioning of shaped cutting edges after their regrindings: by linear displacement of prismatic shaped cutters and angular displacement of round ones respectively

Alexander MIRONENKO

Tatyana TRETYAK

Yury GUTSALENKO

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  Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     164 MODELING OF RUNNING CUTTERS FOR SHAPING OF  IMPROVED NONINVOLUTE TOOTH GEARS    Tatyana TRETYAK1, Yury GUTSALENKO2, Alexander MIRONENKO3  National Technical University “Kharkov Polytechnic Institute”, Ukraine 1,3Associate Professor, 2Senior Staff Scientist 1,3geargroup@ukrmash.com, 2gutsalenko@kpi.kharkov.ua     Abstract:  The  questions  of  tooling  design  for  production  of  advanced  gears  are  considered. Engineering  is  based  on  the  special  applied  development  of  the  mathematical  theory  of  multiparametric mappings of space. In fulfilled engineering of gear cutting tools for shaping of noninvolute gears it is provided for exclusion of distorted profiling after tool regrinds. There are proposed calculation algorithms, which may be used in dataware of respective CAD/CAM systems of maintenance for tooling backup. Among developed tools there are assembled shaping cutters with prismatic and round cutters. Compensatory possibilities of proposed assembled shaping cutters are ensured by repositioning of shaped cutting edges after their regrindings: by linear displacement of prismatic shaped cutters and angular displacement of round ones respectively. Keywords: advanced gearing, constant normal pitch, gear cutter, multiparametric mappings of space  1. INTRODUCTION At the advanced production associations the great attention is given to creation of united information  platforms  and  development  of  simulation  modeling.  Simulation  modeling  is  most successfully applied in tool production as in the high technology field of machining. Proposed in this paper a model of running tools is a set of geometrical and physical-mechanical components. Shaping for such tools is carried out by surface generation method at which work and machine-tool gearings coincide. It allows to increase considerably speed and accuracy of products processing. At resharpening of monolithic shaped gear cutter for processing of noninvolute gears the form of a cutting edge changes. Besides, the form error of tooth gear to be machined occurs due to reduction of center-to-center spacing at displacement of gear cutter. In this connection the problem of development of tools, after resharpening of which noninvolute profile of tooth gear to be machined does not change geometrically, becomes topical. Assembled gear cutters with prismatic and round shaped cutters can be considered as the types of running gear cutting tools for generation of geometry of tooth gear with noninvolute profile. Their advantage is that the form of cutting edges is not deformed when resharpening. Displacement of prismatic cutters by the relevant distance or rotation of round cutters about relevant angle after each resharpening compensate shaped cutting edges displacement relatively gear cutter axis caused by such resharpening. Besides, proposed assembled gear cutters allow quantity of resharpenings several times as much in comparison with the monolithic ones.  2. PROFILING OF ASSEMBLED SHAPING CUTTERS The sequence of shaping of these tools is defined by the fact that gear cutter as running tool is a set of shaped cutters (generating ones in relation to a tool surface and contacting ones in relation to a tooth gear surface to be formed) [1, 2].  The first stage of profiling of assembled gear cutters consists in finding of a tool surface as an envelope. Tooth flank surfaces of gear to be machined and tool surface of the cutting tool are profiled cylindrical ones, they are located in reference points x1y1z1 and x2y2z2 (fig. 1, example for assembled gear cutter with prismatic shaped cutter).    Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     165  Figure 1:To calculation of the profile of assembled shaping cutter with prismatic shaped cutters The  initial  information  at the  first stage  is the  coordinates of points of a profile of processed gear x1К, y1К in gear reference point x1y1z1 and the greatest radius of generating gear R2. Coordinates of points of profile of tool surface x1И, y1И in gear cutter reference point x2y2z2 are unknown quantities. The algorithm of calculation of enveloping surfaces  for running tools and processed tooth gears can  be used to find a profile of tool surface conjugated to the predetermined profile of the processed tooth gear [3]. The second stage of profiling of assembled gear cutters consists in a finding of a shaped cutting  edge  as  a  line  of  crossing  of  tool  surface  and  rake  face.  The  initial  information  is coordinates of points of profile of cylindrical tool surface in reference point of gear cutter x2y2z2 and rake  . Coordinates of shaped cutting edge in the reference point of cutter, marked in fig. as x3y3z3, are unknown quantities. The cylindrical tool surface is formed by action of the operator of parallel shift    on   i ts  p r of i l e .   I n   r e f e r e n c e  p oi n t  of   g e a r   c u t t e r   x 2y2z2 its equation has the following operator and matrix notations with parameter   : , r r И 2 2                  ( 1)  , m m mИ 2 2 r r                ( 2)  wh er e    . 00m ;xyxm ;xyxmИ 2И 2И 2r222r И 2 2         (3) Th e rake  f ac e (p l a n e)  i s  f o r m ed  by  act i o n  o f  t w o  operato rs o f  para l l e l  shif t   1   an d   2   o n b a s e  po i n t   o f   cutt i n g  edge  w i t h   coo r di n at es   x 2б, y2б, z2б [2]. Having directed the vectors of carryover as shown in figure, we will write down the operator and matrix equation of face plane in reference point of gear cutter x2y2z2with parameters  1   an d   2  :   Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     166 , r r 2 1 б 2 2                ( 4)                                                                              , m m m m2 1 б 2 2 r r                                                                                          ( 5)                                        wh er e    .sin0cosm ;00m ;zyxm221б 2б 2б 2r 2 1 б 2                                             (6 )  F o r asse m bl ed gear cutt er wi t h  pri sm at i c sh aped c utt ers ; R c x 2 23 б 2       ; 0 y б 2      , 0 z б 2           ( 7)  an d  f o r   as s e m b l ed  gear   cutt er   wi t h   r o un d  s h aped  c utt er s ; cos R c x 23 б 2         ; 0 y б 2      , sin R zб 2           ( 8)  c23– center-to-center spacing of reference points x2y2z2 and x3y3z3,  – clearance angle, R – the biggest radius of tool. Having equated the right parts of the equations (2) and (5), we obtain the condition of crossing of tool surface and rake face in matrix notation: . m m m m m2 1 б 2 И 2 r r                ( 9)  T hi s   co n d i t i o n   i nc l ude s   t h r ee  equat i o n s   w i t h   p ar a m et er s   o f   t oo l   s ur f ac e  a n d  r ake  f ace . T h e i r   s o l ut i o n   m ake s   po s s i bl e  t o   def i ne  un k n o wn   par a m et er s  ,  1  ,  2  ,   an d  t h en   by   m ea ns  o f   t h e  equat i o n   ( 5)   to   def i ne  co o r di n at es   o f   po i n t s   o f   cut t i n g  edge  X2, Y2 , Z2  in reference point of gear cutter x2y2z2. For implementation of next stage of profiling it is necessary to write down coordinates of points of shaped cutting edge in cutter reference point x3y3z3. The operator and matrix equations of transition from reference point of gear cutter x2y2z2 to cutter reference point x3y3z3 by means of coordinate operator  23 c  wri t e do wn  as f o l l o ws :  , с r r 23 РК 2 РК 3              ( 10)  , m m m23 РК 2 РК 3 c r r             ( 11)  wh er e    .cos0sinm333             (12) Th e   th i r d   s ta g e   of   p r of i l i n g   of   a s s em b l e d   g e a r   c u t te r s   c on s i s ts   i n   th e   a n al y t i c al   d e s c ri p ti on   o f  s h a p e d  b a c k  s u r f a c e  of  r o ta ti on  i n  r e f e r e n c e  p oi n t of  c u t te r  a n d  a  f i n di n g  of  p r of i l e  of  th i s  s u rf a c e   i n  s ta n d a r d  c r os s -s e c ti on . Th e  i n i t i al  i n f orm a ti on  a t  th i s s ta g e  i s  c o or d i n a te s   of  p oi n ts  of  s h a p e d  c u t ti n g e d g e ,   t o ol  c l e a r a n c e   , and also the maximum radius R for round shaped cutter. The  cylindrical  flank  surface  of  prismatic  shaped  cutter  can  be  formed  by  action  of  the operator of parallel shift    on   s h a p e d   c u t ti n g   e d g e   [ 2 ] .   I ts   e q u a ti on s   i n   op e r a t o r   a n d   m a tr i x   n o ta ti on  a r e   a s   f ol l ow s :  , r r 3 РК 3 3              ( 1 3 )  , m m m3 РК 3 3 r r              ( 1 4 )  wh er e    .cos0sinm333             ( 1 5 )  Fl a n k sur f ace o f  ro un d sh aped cut t er can  b e  f o r m e d by  act i o n  o f  t h e rot a t i o n  o perato r    o n   s h aped  cut t i n g  edge  [ 2] .   I t s   equa t i o n s   i n   o per ator   an d  m at r i x   n o t a t i o n   ar e  as   f o l l o w s :      Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     167 , r r РК 3 3               ( 1 6 )  , m m mРК 3 3 r r                             ( 1 7 )  wh ere    .cos 0 sin0 1 0sin 0 cosm             ( 1 8 )  Let ' s  draw  an   a xi a l   p l a n e  o f   st an d ard  cro ss -sect i o n   N -N.  I t s  o pera to r   an d  m at rix equat i o ns w i t h  para m et ers x3Н and y3Н in cutter reference point x3y3z3  write down as follows: , r r Н 3 3              ( 1 9 )  , m mH 3 3 r r              ( 2 0 )  wh er e    .0yxm H 3H 3r H 3          ( 2 1 )  Havi ng  equat ed  t h e  ri g h t   par t s   o f   t h e  equat i o ns   (14)  an d  (20)  we  wi l l   o b t a i n   t h e  co n d i t i o n   o f   cro s s i ng  o f   fl a n k   s ur f a ce  o f   pr i s m at i c  s h aped   cut t er  an d  s t an dard  cro s s -s ect i o n p l a n e  i n   m at r i x   n o t a t i o n :  . m m mH 3 3 РК 3 r r              (22) Thi s   co n d i t i o n   i nc l udes   t h ree  equ at i o ns   w i t h   par a m et ers   o f   fl a n k  s ur f ace  a n d  s t an dard  cro s s -s ect i o n   p l a n e.   Th e i r  s o l ut i o n   a ll o ws   t o  def i n e  un k n o wn   para m et ers 3  ,   x 3Н and y3Н and connected with x3Н parameter z3Н, and hereby to find coordinates of points of required profile of flank surface of prismatic shaped cutter in standard cross-section. Having  equated  the  right  parts  of  the  equations  (17)  and  (20)  we  will  obtain  the condition of crossing of flank surface of round shaped cutter and standard cross-section plane in matrix notation: .3P 3H r K r m = m m             (23) It includes three equations with parameters of flank surface and standard c ross-section plane. Their solution allows to define unknown parameters  , x3Н, and y3Н, and thus to find coordinates of points of required profile of flank surface of round shaped cutter in standard cross-section. 3. PHYSICAL-MECHANICAL SIMULATION OF GEAR CUTTER LOADING On the basis of the obtained set of equations, the geometrical model of tool which was a basis for the FEM strength and a heat analysis at processing is developed (fig. 2). The adequacy of FEM model for shaping element with reproduction accuracy of boundary conditions, loading, geometry and properties of material was defined at the first stage of the investigation.  Investigations  on  determining  of  total  displacements  (fig. 3),  equidistant deformation  by  Mises  criterion  and  equidistant  pressure  by  Mises  criterion  (fig. 4)  under condition of pressure application to surface normal, resultant of which is 4000 N (~400 kg) for the materials of high-speed steels group were carried out.   Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     168  Figure 2: Geometrical model of gear cutter (full)       Figure 3: Resultant displacements [m]  Figure 4: Equivalent stress [Pa] Maximal equivalent stress according to Mises criterion, calculated with averaging on nodal point, is =5550 MPa. Given calculated  model  is  not accurate since  it  leads to setting stiffness too high  in comparison with reality which means that it leads to occurring of error too. Distributed load application on the surface allows to decrease the error caused due to application of equivalent force. And with it the direction of distributed load should coincide with axis of cylindrical surface. Taking into account carried out calculations and secondary analysis, the sector model which includes 5 teeth (fig. 5), fixing scheme and distributed load (fig. 6).    Figure 5: Geometrical model of gear cutter sector and boundary conditions Figure 6: FEM model of gear cutter under loading    Fiabilitate si Durabilitate - Fiability & Durability    Supplement no 1/ 2012   Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X     169 A set of experiments are worked out and exacted data array (fragment is presented in table 1), which permit to make conclusion on strength analysis of running gear cutting tool and change of parameters at increase of contact sites, that correspond to tool blunting,  is obtained.  Table 1: Load (F), temperature (T) and strength () FEM calculation simulation  N  tooth F1  [MPa] F2 [MPa] F3 [MPa] F4 [MPa] T1 [C] T2 [C] T3 [C] [MPa] 1  4000  -  -  -  618  -  -  3720 2  4000  -  2000  -  611  -  314  3760 3  4000  3000  2000  -  615  408  308  3800 4. CONCLUSION The optimized geometrical model, simulation FEM model of gear cutter, considering its physical-mechanical properties on the basis of which it is possible to recommend material and geometrical  parameters  of  cutting  elements  for  the  assembled  tool  of  equidistant  tooth generation on noncylindrical surfaces of two-parameter gearing [4] are the findings of the carried out investigations.  REFERENCES [1] Rodin P.R. Fundamentals of shaping of surfaces by cutting. K., Vyshcha shkola, 1977. 192 p. (in Russian) [2] Perepelitsa B.A. Development of the theory of shaping of surfaces by cutting and cutting tools by means of mappings of affine space. D. Sc. Thesis. – Tula, 1981. – 359 p. (in Russian) [3] Kondusova E.B. Tretyak T.E., Krivosheya A.V. Algorithm of calculation of profile of enveloping  surfaces  for  running  tools  and  components.  Proceedings  of  the  fifth International  Conference  “New  Leading-Edge  Technologies  in  Machining  Building”, Rubachie, Ukraine, September 18-21, 1996, рр. 140-141. (in Russian) [4] Gutsalenko Yu., Mironenko A., Tretyak T. Equidistant tooth generation on       noncylindrical surfaces for two-parameter gearing. Fiability & Durability, Targu-      Jiu, CBU, 2011, No. 2(8), pp. 67-72. 

MODELING OF RUNNING CUTTERS FOR SHAPING OF IMPROVED NONINVOLUTE TOOTH GEARS

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