Bodzás, Sándor

Dudás, Illés

University of Debrecen Electronic Archive

DEBRECENI MŰSZAKI KÖZLEMÉNYEK 2010/2 
 49
PRODUCTION GEOMETRICAL ANALYSIS OF LOGARITHM SPIRAL 
BACKWARD TURNED CURVE⊗ 
 
LOGARITMIKUS SPIRÁLIS HÁTRAESZTERGÁLÁSI GÖRBE 
GYÁRTÁSGEOMETRIAI VIZSGÁLATA 
 
BODZÁS, Sándor 1, Dr. DUDÁS, Illés 2 
 
1 PhD student, University of Miskolc, Department of Production Engineering, 3515 Miskolc, Egyetemváros 
         1sandor.bodzas@uni-miskolc.hu 
2 DSc professor, University of Miskolc, Department of Production Engineering, 3515 Miskolc, Egyetemváros  
         2illes.dudas@uni-miskolc.hu 
 
Abstract: Only such tools could be used for high precision machining of shaping surfaces, the profiles of which 
do not change at resharpening. The backward turned hobs are appropriate for this purpose. In case of logarithm 
spiral backward turned curve the profile constancy can be provided in case of the repeated resharpening of hob.  
 
Keywords: hob, logarithm spiral, backward turning  
 
Kivonat: Alakos felületek nagypontosságú megmunkálásához csak olyan szerszámok használhatók, amelyeknek a 
szelvénye újraélezéskor nem változik. Erre a célra a hátraesztergált fogú marók felelnek meg. Logaritmikus 
spirális hátraesztergálási görbe esetén biztosítható a maró többszöri újraélezése esetén is a profilállandóság. 
 
Kulcsszavak: lefejtőmaró, logaritmikus spirális, hátraesztergálás  
 
1. INTRODUCTION 
 
The tiller surface of hob is such worm on which sliver slots are prepared. They backward turn and 
backward grind the hob cogs because of the resharpening assurance. The cog profile of the hob in 
normal plane is equal to the profile of the basic worm (Figure 1). 
 
 
 
Figure 1. Worm gear hobs 
                                                 
⊗  Reviewed paper. Submitted: 29. 09. 2010., Accepted: 22. 10. 2010. 
    Szaklektorált cikk. Leadva: 2010. szeptember 29., Elfogadva: 2010. október 22. 
    Reviewed by Dr. Gábor PÁY / Lektorálta: Dr. PÁY Gábor 
DEBRECENI MŰSZAKI KÖZLEMÉNYEK 2010/2 
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During production the tooth space of the worm gear is equal to hob cog, so the kinematical track of the 
tool creates the worm gear cog profile. This is called direct motion mapping [2, 4]. 
 The advantage of the solution: there is no from cog to cog division, so work is without interruption 
(Figure 2).  
 
 
 
 
Figure 2. Worm gear production 
 
2.  GEOMETRICAL SHAPING OF THE TOOL 
 
We define the limiter surface of one cog of the tool and the cutting edge on Figure 3. The most 
important limiter surfaces of the cog (Figure 3): 
 
- H            face surface;                                 
- RB, RJ     back surfaces (left and right);      
- Fh            back surface of head ribbon;    
- F             head ribbon. 
 
 
 
 
Figure 3. Surface elements and edges of hob  
DEBRECENI MŰSZAKI KÖZLEMÉNYEK 2010/2 
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The most important cutting edges (Figure 3): 
 
-  VB  left side edge as intersection of B tooth and H face surfaces; 
-   VJ  right side edge as intersection of J tooth and H face surfaces; 
-  VF  addendum edge as intersection of F head ribbon and H face surfaces. 
 
 When forming the RB and RJ back surfaces, besides the effect of the chipping process we have to 
consider mainly the effect of the resharpening of the tool (the insurance of profile accuracy). Since the 
hobs have complicated geometry and are expensive tools, it is important to ensure that the tool could 
be resharpened several times when we elaborate the geometry of the tool [1, 3]. 
 
3. DEFINING OF BACKWARD TURNING 
 
A backward turned curve is called that curve in which the back surfaces of the cogs of the hob are 
found. They have to content two conditions in whichever perpendicular section of figure hob axis: 
 
a) The measured profile height in radial direction can not be changed along the curve from the back 
surface. 
b) The α back angle can not be changed in whichever profile point of tool.   
 
 The backward turning is radial directional, the perpendicular section for the hob axis of the back 
surface gives the backward turned curve.  This back surface is prepared by the shaping surface (figure 
knife)  such during workpiece turning it does radial direction motion [2]. Such prepared backward 
motion is repeated in many times as many tool has cog. 
 The backward turning have to be done such curve, in which tangent to curve closes constant  β 
angle with radial direction in one random point of the curve, which suitably the α back angle is 
constant.  This is the logarithm spiral (Figure 4).  
 
 
α
α
 
 
Figure 4. Logarithm spiral 
 
DEBRECENI MŰSZAKI KÖZLEMÉNYEK 2010/2 
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4. MATHEMATICAL ANALYSIS OF LOGARITHM SPIRAL 
 
Given from the structure of the backward turning machine, is created the required shape in polar 
coordinate system, that is certain angle defection appertains to certain radial directional moving, that is 
why we create the equation of the curve in polar coordination shape [4]. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Figure 5. Conceptual figure of logarithm spiral  
 
 Based on well known proposition in mathematics the tangent β angle between the tangent and the 
radius vector can be defined such we compose the quotient of the function and the first differential 
quotient. If rf =)(ϕ , then: 
 
  α
ϕ
β
tg
d
dr
rtg 1==                       (1) 
 
Namely: 
 
  ααβ tgtgtg
1)90( =−= °                     (2) 
 
 Regulated the (1) equation is: 
 
  ϕα dr
drtg ⋅=                         (3) 
 
 If we introduce the αtgm =  nomination and receive α angle constant, which is our condition, 
because it gives the constantly back angle, then == mtgα constant. It is substituted to (3):   
 
  
r
drmd =⋅ϕ                         (4) 
β
β
α
α
ϕ
DEBRECENI MŰSZAKI KÖZLEMÉNYEK 2010/2 
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 After that we integrate both sides:  
 
  ∫∫ =⋅ rdrdm ϕ                        (5) 
 
 Which is in conclusion: 
 
  rcm ln=+⋅ϕ                        (6) 
 
 That is: 
 
  ree cm =⋅⋅ϕ                          (7) 
 
or: 
 
  ϕ⋅⋅= mear                           (8) 
 
 Where  the aec =  or ca =ln (Figure 4). The a is obviously constant value and if 0=ϕ  then 
ra = . Such gained equation is the polar equation of logarithm spiral (Figure 4). Where: 
 
- r – radius vector; 
- a    – constant (if 0=ϕ , then r=a); 
- m – constant exponent ( αtgm = ); 
- α – the angle which is closed by radius vector on one random point and the polar axis on the  
                curve, - constant angle which is closed by the tangent of curve and the perpendicular line  
                for radius vector (back angle).  
 
 The logarithm spiral can be established on the largest distance of the profile form the turning axis.  
The backward turning curve of the whichever farther point of the profile will be such curve, which its 
every point is equal distance from the original logarithm spiral, which is called conchoid of this 
logarithm spiral.  
 The α  angle value changing is such insignificant, after sharpening the hob profile is in tolerance 
level that is why the logarithm spiral can be used.   
 
5. PRODUCTION OF BACKWARD TURNED HOB  
 
The production of the hobs is fairly complex, punctual and precise task. In Figure 6 the different 
production stage of the hob can be seen: 
 
- a  dressing turning; 
- b  flute milling; 
- c  head surface grinding;  
- d  backward turning.   
 
 The backward turning of hob cogs is prepared along logarithm spiral, which is shown by long chain 
in Figure 6. 
DEBRECENI MŰSZAKI KÖZLEMÉNYEK 2010/2 
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Figure 6. Production of backward turned hob  
 
6. DEFINING OF DIMENSION OF BACKWARD TURNING  
 
The most important element in backward turning is defining of the h dimension of backward turning. 
This h dimension characterized the needed radial cam for backward turning [1]. The dimension of 
backward turning is same in one cog pitch and regarded to such cog pitch it can be defined based on 
Figure 7 nominations. Length of arch for one cog pitch on the largest perimeter is:   
 
  
z
Di π⋅=                            (9) 
 
 
 
Figure 7. Defining of dimension of the backward turning  
DEBRECENI MŰSZAKI KÖZLEMÉNYEK 2010/2 
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 Calculation with average α back angle straightening the shaded curve triangle is:  
 
  πα ⋅
⋅==
D
zh
i
htg                          (10) 
 
From this is: 
 
  απ tg
z
Dh ⋅⋅=                           (11) 
 
SUMMARY 
 
Only such tools could be used for high precision machining of shaping surfaces, the profiles of which 
do not change at resharpening or the dimension of profile changes is in the tolerance level. The 
backward turned hobs are appropriate for this purpose.  
 It could be ensured with resharpening along logarithm spiral, profile without deformation could be 
given in axial plane and the back angle of the cogs of the hob is to be constant along the backward 
turned curve. These conditions are indispensable for the high precision, long life working of the tool.  
The main production parameter in backward turning is the h dimension of backward turning. This h 
dimension characterized the needed radial cam for backward turning. That is why defining of h 
dimension is important.  
 
REFERENCES 
 
[1] DR. DUDÁS ILLÉS: Gépgyártástechnológia III. Miskolci Egyetemi Kiadó, 2005 (ISBN 963 661 
572 1) 
[2] DR. DUDÁS ILLÉS: The Theory and Practice of Worm Gear Drives. Penton Press, London, 
2000. (ISBN 1 8571 8027 5) 
[3] DR. DUDÁS ILLÉS: Fogazószerszámok fejlesztésének tendenciái hengeres és kúpos lefejtőmaró    
 gyártásgeometria elmezése, NME, Miskolc 
[4] SASI NAGY ISTVÁN: Fogazószerszámok tervezése, Budapest, Műszaki könyvkiadó, 1961. 
 


Production geometrical analysis of logarithm spiral backward turned curve

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Production geometrical analysis of logarithm spiral backward turned curve

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