Bodzás, Sándor

Dudás, Illés

Analecta Technica Szegedinensia

English

University of Debrecen Electronic Archive

 Analecta Vol. 8, No. 2 ISSN 2064-7964 
45 
 
MATHEMATICAL MODEL TO DETERMINATION OF THE RESHARPENING 
TERRITORY OF CONICAL HOB  
1
I. Dudás, 2S. Bodzás 
1Institute of Manufacturing Science, University of Miskolc, Egyetemváros, 3515, Miskolc, Hungary, 
e-mail: illes.dudas@uni-miskolc.hu 
2Department of Mechanical Engineering, University of Debrecen, Ótemető str. 2-4,4028, Debrecen, Hungary, 
e-mail: bodzassandor@eng.unideb.hu 
 
ABSTRACT 
 
Based on the general mathematical model of Illés Dudás which is appropriate for mathematical modelling of 
production technology methods we have worked out a model for resharpening analysis of conical hob. After the hob 
resharpening using numerical calculations the determination of the tooth surface of face gear by cutting edges is 
necessary for the analysis. Based on this methods we could calculate the permissible critical angle of the hob and the 
profiles of the hob and the face gear in axial section. The permissible critical angle of the hob is the critical angle the 
hob cutting edge of which manufactured face gear profile is situated in the permissible profile error tolerance. We have 
worked out a new geometric conical worm gear drive that is the conical worm gear drive having arched profile. Using 
this mathematical model we have done resharpening analysis for the hob having arched profile and determined the 
permissible critical angle. 
 
Keywords: hob, face gear, mathematical model, resharpening 
 
1. INTRODUCTION 
 
The teething of a face gear is determined by the teething of the worm with contact surfaces mutually 
meshing with each other.  
The teething of the worm, such a tool should be applied so that its cutting edges are situated on the 
surface of a substituting worm (Fig. 1) [3]. It is necessary so that the production tolerances could be 
provided.  
 

 
 
Figure 1. The basic profile of the face gear hob 
 
The elements of the face gear hob is equal with the elements of the worm co-operating as a gear drive 
with the face gear by direct motion mapping [3]. Tip diameter and tooth thickness of the hob is larger than 
the parameters of the real worm because of the assurance of resharpening territory and backlash between 
the worm and the face gear (Fig. 1).  
Based on these it could be determinable the equation of the hob cutting edge has to be known to 
determination of the tooth surface points of the spiroid face gear [2].  
 
 
 Analecta Vol. 8, No. 2 ISSN 2064-7964 
46 
 
2. DEFINING OF THE HOB CUTTING EDGE AND THE FACE SURFACE OF THE HOB  
 
Given the   ,,gr

 generator curve of the B helicoidal surface having parameters pa (axial) and pr 
(radial thread) in the form (Fig. 2) 
 
  gszFF rMr

 ,11 ,              (1) 
 
      
 
 
  





aF
rF
F
pz
py
x
1
1
1
cossin
sincos
                  (2) 
 
which could be random profile (line, circle, elipse, parabola, etc.). 
 






 
 
Figure 2. Defining of conical helical surface 
 
 
 
Figure 3. The generation of face surface H and cutting edges V [3] 
 
To face surface H is a ruled (closed and flat) helicoidal surface with a generator line perpendicular to 
the axis of surface B showing a lead direction perpendicular to the lead direction measured on the pitch 
cone of helicoid B (Fig. 3) 
 
   ,11
H
F
H
F rr

  .            (3) 
 Analecta Vol. 8, No. 2 ISSN 2064-7964 
47 
 
The common solution of the equations of the worm surface (B) and the face surface (H) result the 
profile points of the cutting edge of the hob (V) (Fig. 3) [3], that is  
 
           ,11 FF rr

             (4)  
         ,1 VFr
  .                   
  ,11
H
F
H
F rr

        
 
Let the parameter of the relief thread surface be '
ap  based on (2) the equations of the back surface of 
hob are: 
 
        
 
 
  





'
cossin
sincos
ahr
rhr
hr
pz
py
x
  .                       (5) 
 
3. MATHEMATICAL MODEL TO DETERMINATION OF THE TOOTH SURFACE POINTS OF 
FACE GEAR DURING HOB RESHARPENING 
 
Knowing the  111 ,
V
F
V
F rr

 two parametric vector scalar function of the hob cutting edge (meshing 
surface) our objective is to generate the tooth surface points of the face gear during resharpening along the 
face surface of the hob. 
 
 
 
Figure 4. The position of spatial coordinate systems to determination of the tooth surface points of the face gear 
 
For the description of the motion relations we define the own motion of each coordinate system (Fig. 
4). Thus K1F (x1F, y1F, z1F) coordinate system rotates with 
 
             .11 const
dt
d



                                     (6) 
 
angular velocity in K1cs (x1cs, y1cs, z1cs) stationary coordinate system [3, 4, 5].  
 
 Analecta Vol. 8, No. 2 ISSN 2064-7964 
48 
 
The K2F (x2F, y2F, z2F) coordinate system in the K2 (x2, y2, z2) coordinate system rotates with 
 
             .22 const
dt
d



               (7) 
 
angular velocity [3, 4, 5]. 
Transformation matrix between the K1F (x1F, y1F, z1F) rotational coordinate system fixed to member 1 
and the K2F (x2F, y2F, z2F) rotational coordinate system fixed to member 2 (Fig. 4) is: 
 
 FcscsFFF MMMM 1,11,22,21,2  
 
                        















1000
0cossin
cossincossinsincossin
sincossinsincoscoscos
11
2222112
2221212
c
ba
ba



    .              (8) 
 
3.1. Application of the direct kinematical method to generation of the tooth surface points of the face 
gear  
 
Given the  111 ,
V
F
V
F rr

 two parametric vector – scalar function of the hob cutting edge in the K1F (x1F, 
y1F, z1F) rotational coordinate system of the hob [2]. 
In the K1F coordinate system the normal vector could be calculated [3, 5]: 
 
   
1
11
1
 





V
F
V
Fv
F
rr
n

             (9) 
 
The relative velocity between the two surfaces can be defined by the transformation between the K1F 
rotational coordinate system of the hob and the K2F rotational coordinate system of the face gear in the K1F 
system: 
 
        
F
FF
FFF r
dt
dM
Mv 1
1,2
2,1
)12(
1

                  (10) 
where 
           FFFFk M
dt
d
MP 1,22,11 
                                      (11) 
 
matrix of the kinematic motion mapping:  
 
          
















0000
0sincos
sinsin01
coscos10
21121121
121121
121121
1
iaii
ibi
ibi
P



.
                      (12) 
 
Knowing the Connection I. Statement and the equation of the hob cutting edge (4) the tooth surface 
points of the face gear by cutting edges are the common solution of the below equations:  
 
 111 ,
V
F
V
F rr

 
                          0)12(11 
V
F
V
F vn

         .        (13) 
  V
FFFF rMr 11,22

  
 
 Analecta Vol. 8, No. 2 ISSN 2064-7964 
49 
 
Using the received equations (13) for example we have done resharpening analysis for the our worked 
out, new geometric conical worm gear drive having arched profile in axial section [1]. On Fig. 5 it could be 
seen the hob face surface is situated in the axial section.  
We have done computer program to the analysis (Fig. 6). As a result considering of the profile 
preciseness of the face gear we have received the permissible critical angle of the hob υhat=5° (Fig. 5 and 
6). 
 



 
 
Figure 5. Definition of the permissible resharpening provision and the resharpening critical angle of the hob  
 
    
       a) υ=0°                       b) υ=5° 
 
Figure 6. The geometric establishments of hob cutting edges 
 
4. CONCLUSION 
 
Based on the general mathematical model of Illés Dudás [3] we have worked out a model to 
determination of the tooth surface points of the face gear which are generated by the hob cutting edges in 
discreet resharpening angle position.  
Based on the permissible profile tolerances of the face gear and the hob the resharpening critical angle 
of the hob could be determined in the hob axial section.  
This is the critical angle the hob cutting edge of which manufactured face gear profile is situated in the 
permissible profile error tolerance. Consequently it is correct in case of operating.    
For example we have shown in our results for the hob cutting edge on Fig. 6. 
 
 
 
 
 Analecta Vol. 8, No. 2 ISSN 2064-7964 
50 
 
ACKNOWLEDGEMENT 
 
The development of the production technology and the real production process of the conical worm 
gear drive and the hob having arched profile in axial section are occurred in the DifiCAD Engineering 
Office Ltd. (Miskolc, Szentpéteri Gate 5-7.) (Fig. 7). Director: Dr. Illés Dudás. 
 
 
 
Figure 7. The hob backward grinding could be seen on this figure (DifiCAD Engineering Office Ltd.) 
 
REFERENCES 
 
[1] Dudás, I., Bodzás, S., Dudás, I. Sz., Mándy, Z. (2014): Konkáv menetprofilú spiroid csigahajtópár 
és eljárás annak köszörüléssel történő előállítására, Szabadalmi bejelentés napja: 2012.07.04., 
Szabadalmi lajstromszám: 229 818 
[2] Bodzás, S. (2014): Kúpos csiga-, tányérkerék- és szerszám felületek kapcsolódásának elemzése, 
Ph.D. értekezés, Miskolci Egyetem, p. 154., Doktori témavezető: Prof. Dr. Dudás Illés, DOI 
10.14750/ME.2014.006 
[3] Dudás, I. (2004): The Theory and Practice of Worm Gear Drives. Kogan Page US., USA 
[4] Hegyháti, J. (1988): Untersuchungen zur Anwendung von Spiroidgetrieben. Dissertation, TU 
Dresden  
[5] Litvin, F. L., Fuentes, A. (2004): Gear Geometry and Applied Theory, Cambridge University 
Press, ISBN 978 0 521 81517 8 


Mathematical model to determination of the resharpening territory of conical hob

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Mathematical model to determination of the resharpening territory of conical hob

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