Bodzás, Sándor

Analecta Technica Szegedinensia

English

University of Debrecen Electronic Archive

Vol. 9  No. 1 
 
1 
 
COMPUTER AIDED GEOMETRIC MODELLING OF CYLINDRICAL WORM 
GEAR DRIVE HAVING ARCHED PROFILE 
S. Bodzás 
Department of Mechanical Engineering, University of Debrecen, H-4028 Debrecen, Ótemető str. 2-4. 
e-mail: bodzassandor@eng.unideb.hu 
 
ABSTRACT 
 
The objective of this publication is creation of computer geometric models (CAD) of developed in Diósgyőri Machine 
Factory and using exact production geometry first produced (grinded) cylindrical worm gear drive having arched 
profile by Illés Dudás [3, 4, 5] for connection analysis, Finite Element Method analysis, etc.  
Since the worm and worm gear wrap each other, that is why the worm gear has to be produced by a tool geometry of 
which is similar to the worm geometry. Knowing of the Connection I. Statement tooth surface of driven element which 
is wrapped by driver element is defined by numerical calculations. 
The CAD model of the worm gear could be designed by adaptation of interpolating B – spline surface to the tooth 
surface points of worm gear.  
  
Keywords: worm, worm gear, hob, CAD, modelling 
 
1. INTRODUCTION 
 
One the most modern types of cylindrical helicoidal surfaces is the worm generated using a circular profile 
tool. 
Contact surfaces between worms having ruled surfaces (Archimedian, convolute, involute types) and 
mated worm gears do not allow the formation of a continuous, high pressure bearing oil film. It is best to 
build up an oil film between mated surfaces so that the direction of the relative velocity of the drive faces 
into the direction normal to the common contact curve or very close to it [2, 3, 4]. 
 
 
Figure 1. Cylindrical worm having arched profile [3] 
 
A further advantage of mated elements having curved profile is that radius of mating flank surfaces are 
situated on the same side of the contact point common tangent, that is concave and convex surfaces are in 
contact, generating relatively small Hertz stress [2, 8]. 
As a consequence of smaller contact pressure, the load-carrying oil film can form easier. 
Vol. 9  No. 1 
 
2 
 
As a result of arched profile teething the tooth shape and the suitable positioning of the centre of curvature 
of tooth flank (the position of engagement line) the dedendum tooth thicknesses, both on the worm 
1FS  and 
worm gear 
2FS , are significantly wider [2, 6]. 
 
2. DEFINING OF COORDINATE SYSTEMS 
 
The Illés Dudás type general mathematical model which is appropriate for mathematical modelling of 
production technologies methods is considered for defining of the necessary coordinate systems for 
modelling [3, 5].  
 
ϕ
ϕ ω
ϕ
ϕ
ω
 
 
Figure 2. Coordinate systems dispositions for modelling of cylindrical worm gear drive  
 
For the description of the motion relations we define the own motion of each coordinate system. Thus 
K1F (x1F, y1F, z1F) coordinate system rotates with 
 
.11 constdt
d
==
ϕω
  (1) 
 
angular velocity in K1cs (x1cs, y1cs, z1cs) stationary coordinate system [3, 5, 7].  
The K2F (x2F, y2F, z2F) coordinate system in the K2 (x2, y2, z2) coordinate system rotates with 
 
.22 constdt
d
==
ϕ
ω
  (2) 
 
angular velocity [3, 5, 7]. 
Vol. 9  No. 1 
 
3 
 
Transformation matrices 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  are (Fig. 2): 
 
=⋅⋅= FFFF MMMM 1,11,22,21,2  












−
⋅−⋅⋅
⋅⋅⋅
=
1000
00sincos
cossincoscoscossin
sincoscossinsinsin
11
221221
221212
ϕϕ
ϕϕϕϕϕϕ
ϕϕϕϕϕϕ
a
a
 (3) 
 
=⋅⋅= FFFF MMMM 2,22,11,12,1  












−
⋅−⋅⋅
⋅−−⋅⋅
=
1000
00sincos
cossincoscossincos
sincoscossinsinsin
22
112121
112112
ϕϕ
ϕϕϕϕϕϕ
ϕϕϕϕϕϕ
a
a
 (4) 
 
3. GENERATION OF THE TOOTH SURFACE POINTS OF WORM GEAR 
 
The dentation of worm gear is produced by hob which has similar geometry to worm connecting with 
worm gear based on the direct motion mapping [1, 2, 3, 4]. 
That is why we have to know the two parametric equations of the cutting edge of hob for modelling [3, 4]: 
 
( )221
1
1
cos
sin
ηρϑ
ϑη
ϑη
−−−⋅=
⋅=
⋅−=
Kpz
y
x
axF
F
F
 
(5) 
 
The tooth surface points of worm gear are determined by direct kinematic method. Knowing the cutting 
edge of hob and the Connection I. Statement we search the tooth surface points of worm gear which are 
generated by mutual wrapping.  
The relative velocity vector is in the K1F  system: 
 
FkF
FF
FFF rPrdt
dM
Mv 111
1,2
2,1
)12(
1
 ⋅=⋅⋅=  . (6) 
 
Based on (6) the derived matrix is: 
 


























⋅⋅−⋅−
⋅−
⋅⋅−
⋅⋅−
⋅
⋅⋅⋅−
⋅−
⋅⋅
⋅+
⋅⋅
=
0000
00cossin
sincos
sincos
cossin
sinsin
coscos
cossin
sinsin
coscos
cossin
sincos
11
221221
12
1221
1221
21
221221
12
1221
12
1221
1,2
ϕϕ
ϕϕ
ϕϕ
ϕϕ
ϕϕ
ϕϕ
ϕϕ
ϕϕ
ϕϕ
ϕϕ
ϕϕ
iai
i
i
iai
ii
M
dt
d
FF
 
(7) 
 
Vol. 9  No. 1 
 
4 
 
The P1k matrix of the kinematic motion mapping is: 
 












⋅⋅⋅
⋅−
⋅−−
=
0000
0cossin
0cos01
0sin10
21121121
121
121
1 iaii
i
i
Pk ϕϕ
ϕ
ϕ
 (8) 
 
If η and ϑ parameters are independent from one another, then the normal vector can be calculated by the 
following way [2, 5]: 
 
ϑη ∂
∂
×
∂
∂
= FFF
rrn 111

  (9) 
 
The equations of the tooth surface of member 2 which can be defined as the mashing surface of the group 
of contact lines in the K2F coordinate system [3, 7]: 
 
0)12(11 =⋅ FF vn

 
),(11 ϑηFF rr

=  
FFFF rMr 11,22

⋅=  
(10) 
 
Knowing of the (10) equations we have worked out a computer program in case of a concrete cylindrical 
worm gear drive [2, 3, 5] for producing of the tooth surface points of worm gear (Fig. 3). 
On Fig. 3 it could be seen the touched tooth surface points by the cutting edge of hob namely the tooth 
surface points of worm gear in the y2F – z2F plane.  
 
 
 
Figure 3. Determination of tooth surface points of worm gear  
 
 
 
Vol. 9  No. 1 
 
5 
 
4. COMPUTER AIDED MODELLING 
 
The generated TXT file by computer program is imported to Solid Works designer software.  
On the profile points interpolating B spline was drawn, then tooth profile, along the circumference of the 
worm gear, was determined according to number of the gear cogs [1]. Limiting of the dentation territory 
has been occurred based on geometric data and technical drawing of the worm gear drive [2, 3]. 
 
 
 
a) Connection cylindrical worm – worm gear model 
 
 
 
b) Connection cylindrical hob – worm gear model 
 
Figure 4. CAD models of cylindrical worm, hob and worm gear 
 
Vol. 9  No. 1 
 
6 
 
Using the stated mathematical method for determination of the tooth surface points of worm gear, the 
worked out computer program and the Solid Works designer software the CAD models of cylindrical 
worm gear drive (worm, worm gear and hob) could be produced (Fig. 4). 
 
5. CONCLUSION 
 
The tooth surface of worm gear is determined by the worm dentation based on mutual wrapping. That is 
why in case of toothing the worm gear has to be produced by hob which has similar geometry to worm.  
Knowing the cutting edges of the hob we have determined the wrapped surface by the hob by numerical 
way namely the tooth surface of the worm gear. 
We have connected interpolating B – spline surface for the tooth surface points of worm gear and after 
produced the CAD models of cylindrical worm, worm gear and hob.  
The received CAD model is suitable for other Finite Element Analysis and connection analysis.  
 
REFERENCES 
 
[1] Bodzás, S.: Kúpos csiga-, tányérkerék- és szerszám felületek kapcsolódásának elemzése, Ph.D. 
dissertation, University of Miskolc, 2014., p. 154. 
[2] Dudás, I.: Ívelt profilú csigahajtás egyszerűsített gyártása és minősítése, University doctoral 
dissertation, Miskolc, 1973. 
[3] Dudás, I.: The Theory & Practice of Worm Gear Drives. Kogan Page US, Sterling, USA, 2004. 
[4] Dudás, I.: Ívelt profilú csigahajtások szerszámozásának és gyártásának fejlesztése, Candidate 
dissertation, Miskolc, 1980. p.153+30 attachment 
[5] Dudás, I.: Csavarfelületek gyártásának elmélete. Academic doctoral dissertation, Miskolc, 1991. 
[6] Krivenko, I. SZ.: Novüe tipü cservjacsnüh peredacs na szudah, Izd. Szudoszrovenie, Leningrád, 
1967. 
[7] Litvin, F. L., Fuentes, A.: Gear Geometry and Applied Theory, Cambridge University Press, 2004. 
[8] Niemann, G.: Maschienenelemente. Band.2., Sprienger-Verlag, Berlin, Heidelberg, New York, 
Tokyo, 1965. 


Computer Aided Geometric Modelling of Cylindrical Worm Gear Drive Having Arched Profile

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Computer Aided Geometric Modelling of Cylindrical Worm Gear Drive Having Arched Profile

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