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CONSIDERATIONS ON THE
GEOMETRICAL ELEMENTS
CALCULATED FOR CIRCULAR ARC
TEETH BEVEL GEARS, 528 SARATOV
TYPE
Niculae GRIGORE
Adrian CREITARU
Abstract: The work presents theoretical and
technological considerations regarding the circular arc
bevel gear type 528 SARATOV, technological principles
developed for machining of this type of teeth, manner of
choosing it and construction of tooth by means of cutting
tools used with this method.
The work approaches the algorithm of calculation for
geometric elements of the teeth and specific parameters of
construction and control for cutter holders used in thetooth
construction of such gears.
Key words: bevel gear, circular arc teeth, Saratov gear
cutting machine, gear geometric elements
1. GENERAL CONSIDERATIONS
The circular arc teeth bevel gears are used for 3...40 m/s
speed range conditions [2], [7].At higher speed conditions, the
teeth shall be ground afterthermal treatment and curved tooth bevel
gears have got
the following advantages: silent operation; large contact ratio;
long lasting in operation; allowing for high gearing (velocity)
ratios; low overall etc.The principle at the basis of the curved
teeth bevel gearmachining consists in generating, by an
imaginarycrown(plain) wheel pattern, a tooth construction tool to
get eachsingle tooth of such face gear (Fig.1) [1], [7], [9].
The cutting tool for the tooth construction of such type ofbevel
gears is a milling cutter head on which externalcutters are
fastened that cut the external side of the cutter
head while inner cutting tools cut the internal side of
thecutting tool head.The cutter head carries out the main rotation
motion with
at the same time generating the tooth of the plain wheel.
Fig. 1. Crown(plain) generating wheel pattern
2. TECHNOLOGICAL ASPECTS
In order to achieve the profile of the bevel gear there also
must be, during tooth construction, a rolling motionbetween the
gear and the generating plain wheel [3], [7].The shape of a cutter
head (holder) and details over cutter
holders are shown in figure 2.
Fig. 2. Cutter head (holder) assembly
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Nominal diameters of the main cutter holder heads areshown in
table 1.
Table 1. Nominal Diameters of Cutter Holder Heads
Ds[in] 3 1/2 6 9 12 18
Ds[mm] 88.9 152.4 228.6 304.8 457.2
Selection of a certain size of the cutter head will be done
depending on the gear modulus, mtand the length of its
conedistance (R, element of the cone) [7], [10].To choose the
typo-dimension of the tooth constructionhead, such nomographic
chartare used as the type of that
shown in figure 3.Typically, in order to machine bevel gears
with curvedteeth and constant height, the unilateral method is
usedconsisting in the fact that in machine tooth finishing onboth
gears (rack and pinion) cutting of concave andconvex parts is
separately achieved.
To roughen the two components of the gear, a same head
of gear cutter head is bilaterally used.Convex parts of the
pinion teeth are achieved by means ofinner cutters while concave
parts are achieved by meansof external cutters of the cutter holder
head.
After teeth has been rough-machined, the finishing job iscarried
out for convex sides by changing coordinates onthe cutter holder
head and then finishing of concavessides, by properly changing
again the head coordinates.
Fig. 3. Nomographic chart used for Selection of the Size
of the Cutter Head proper to the Bevel Gear Tooth
This method of machining curved bevel teeth is used incase of
small scale production. This way, a favourablearea is assured in
contact gearing between conjugated
sides of the gears.
3. CALCULATION OF GEOMETRICAL
ELEMENTS OF 528 SARATOV CIRCULAR
ARC TEETH CONSTANT HEIGHT BEVEL
GEARS
Further on, in figure 4, you have the computing algorithmfor the
gear geometric elements [7].
Fig. 4. Basic rack tooth profile and the constant height
circular arc teeth bevel gear assembly
3.1. Basic data
The teeth numbers of the gears are done by the topic:
on the pinion:z1; on the gear:z2.Outside module (frontal)
is:
m
Gear ratio, u:
1
2
z
zu= (1)
Medium inclination pitch angle:
m
Pressure pitch (normal) angle:
020=n (2)
Reference tooth addendum coeficient, *ah :
0.1* =ah (3)
Reference dedendum clearance coeficient, *c :
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25.0* =c (4)
Face width coeficient:
4...31
===b
Rk
R
b
(5)
Radial profile displacement coeficients:
on the pinion:
=
2
11cos49.0
1 uxr (6)
- on the gear:12 rr
xx = (7)
Tangential profile displacement coeficients:
on the pinion, it must be chosen related to gear ratio:Table 2.
Recomandations for tangential profile
displacement coeficients choice
u 1...2 2...2.5 2.5...3 >3
1tx 0 0.16 0.17 0.18
on the gear:12 tt
xx = (8)
3.2. Calculation of the bevel gears geometrical
elements
Pitch angle:
on the pinion:
=
u
1arctg1 (9)
on the gear:( )uarctg2 = (10)
Pitch diameters:
on the pinion:11 zmd = (11)
on the gear:22 zmd = (12)
Outer cone distance:
2
2
1
1
sin2sin2
ddR == (13)
Face width:
Rk
Rb R
b
== (14)
Mean cone distance:
2
b
RRm =
(15)
Inner cone distance:
bRRi = (16)
Module (interior):
b
bi
k
kmm
1= (17)
The addendum:
on the pinion:( ) iraa mxhh += 11
*
, cu 21 rr xx = (18)
on the gear:( ) iraa mxhh += 22
* , cu12 rr
xx = (19)
The dedendum:
on the pinion:( ) iraf mxchh += 11
** , cu21 rr
xx = (20)
on the gear:( ) iraf mxchh += 22
** , cu12 rr
xx = (21)
The whole depth of teeth:
( ) ia mchh += **2 , unde hhh == 21 (22)
Outside (addendum) diameters:
on the pinion:11 cos2 11 aa hdd += (23)
on the gear:22 cos2 22 aa hdd += (24)
Root (dedendum) diameters:
on the pinion:11 cos2 11 ff hdd = (25)
on the gear:22 cos2 22 ff hdd = (26)
Addendum cone angle:
on the pinion:0
1=a (27)
on the gear:02 =a (28)
Dedendum cone angle:
on the pinion:0
1=f (29)
on the gear:02=f (30)
Outer addendum angle:
on the pinion:1
1
=a (31)
on the gear:22
=a (32)
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Inner dedendum angle:
on the pinion:11 =f (33)
on the gear:22
=f (34)
Distances from the apex of the pitch cone to the back of
the hub:
on the pinion:11 sincos 11 = aa hRH (35)
on the gear:222 sincos 2 = aa hRH (36)
Mounting distances:
1L and 2L shall be chosen by constructive necesities.
Addendum distances:
on the pinion:11 1 aa
HLL = (37)
on the gear:22 2 aa
HLL = (38)
The nominal diameter of the cutter holder, sD , shall be
chosen out of figure 3, depending onRand m.The arc bevel teeth
angle of splitting slope is variablealong the flanks of gear (fig.
5). Therefore in order todefine the indexing slope external angle,
the outside of the
tooth arc is considered a radial direction tangent to sucharc
(point A), while for the indexing slope internal angle
inside the arc, a radial line tangent into point B.
Similarly,the medium indexing slope angle can be defined into
apoint located at half the width of the gear teeth (point M).
Fig. 5. Definition of external (e), internal (i) andmedium (m)
indexing slope angles
The external indexing slope angle, e, may be determinedwith the
relation:
sb
b
b
be
D
R
k
k
k
k
+
=
2
2
121sin
2
12sin (39)
The graphical method [4], [5], [6], [7] allows for
quickdetermination of such angle by making use of the
nomogram presented in figure6.
Fig. 6. Nomogram used to determine the external
indexing slope angle (e)
The internal indexing slope anglei, may be
analyticallydetermined with relation:
( ) ( ) bsbb
b
bi
kDR
kk
kk
+
=
1443sin
1212sin (40)
For the quick graphical determination of the internalindexing
slope angle (i), there is the nomogram given infigure 7 [5], [6],
[7].
Fig. 7. Nomogram used to determine the internal indexing
slope angle (i)
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3.3. Calculation of some parameters of the cutter
holder head [7]
The nominal diameter of the cutter holder, sD , is to be
chosen out of the nomogram shown in figure 3.The nominal radius
of the cutter holder head is:
2
s
s
D
r =
(41)
Eccentricity of the cutter head axis:
msmmsS rRRrOOe sin222 +== (42)
Number of the teeth of the reference face gear is:
2
2
2
10 zzz += (43)
Shifting of tool points of cutter head for milling, if
finishing the gear:
inicrmW
= 13.0tg5.2cos
2
(44)
Actual shifting of the tool points of cutter at the head
formilling, if finishing the gear:
= crr WW , (45)
Rounding has to be done until the value that is the closestto
the normalised value; the positive value will not be
over 0.02mi. Shifting of tool points of cutter at the headfor
milling, if roughing out in the gear:
rer WW = (46)
Shifting of tool points of cutter at the head for milling,
ifroughing out in the pinion:
rep WW = (47)
Shifting of tool points of cutter at the head for milling,
iffinishing in the pinion:
rp WW = (48)
3.4. Calculation of control elements for circular
arc teeth of constant height, model 528
SARATOV
Frontal pitch tooth thickness:
on the pinion:
ecos
tg2
2
1
1
nirt
mxms
+
= (49)
on the gear:12 tt
sms = (50)
Intermediate coefficient:
eeG cossin2
12 = (51)
Decreasing coefficient of the tooth:
on the pinion:21
11 GR
sK
t= (52)
on the gear:22
21 GR
sK
t= (53)
Central semi-angle corresponding to the normal
tooththickness:
on the pinion:1
3
1
1 coscos1 et
d
s= (54)
on the gear:2
3
2
2 coscos2 e
t
d
s= (55)
Design coefficients:
on the pinion:6
1sin 21
11
=K i
4
cos1 121
=K (56)
on the gear:
61
2
212
=K i
4
222
=K (57)
Tooth thickness measured on constant span at
externalextremity:
on the pinion:etcn KsKs cos111 11 = (58)
on the gear:etcn KsKs cos212 22 = (59)
Sharpening of the tooth (deviation of the tooth thickness):
on the pinion:etsKh cos1211 = (60)
on the gear:etsKh cos2222 = (61)
Height measured on the constant span:
on the pinion:1111
hKhh acn += (62)
on the gear:2222
hKhh acn += (63)
4. CONCLUSIONS
The work presents the calculation of the main
geometricalelements of bevel gears with circular arc teeth gears
andconstant height of the teeth, type 528 SARATOV. Theabove
presented lead to the following main conclusions:
For this type of gears it is necessary to be specified:
theinitial data, calculation of geometrical elements of thegear and
clutch, calculation of some parameters of thecutter holders for
teeth manufacturing as well ascalculation of control elements for
circular arc teeth.
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Some of the advantages in the use of this type of bevelgear are
highlighted: much smoother tooth action,increase of the gear
durability, increase of the facecontact ratio versus straight bevel
teeth gears,possibility to achieve bevel gears with higher
velocityratios etc..
In teeth gear manufacturing a significant aspectrelated to the
positional adjustment of the cutter holderin view of assuring the
angles of external indexing
slope (e) and internal indexing slope (i) whichwere determined
by analytical or graphical way.
In the case of these kind of bevel gears the specific ofthe
curved teeth of constant height brings in operation
a significant contribution by equalizing the teethloading,
especially on the top side of gear bevels.
The work is extremely useful for specialists proposingthemselves
to re-design straight teeth bevel gears toreplace such with bevel
gears having circular arc teeth.
REFERENCES
[1] CHISIU, Al., a. o., Organe de masini, EdituraDidactica si
Pedagogica, Bucuresti, 1981, pp 579-586
[2] GAFITANU, M., a. o., Organe de masini, vol. II,Editura
Tehnica, Bucuresti, 1983, pp 278-330[3] GRAMESCU, T., Tehnologii de
danturare a rotilor
dintate, Editura Universitas, Chisinau, 1993, pp 188-197
[4] GRIGORE, N. a. o. Metoda grafica pentru
determinarea unghiului de inclinare de divizare
exterior al danturii rotilor dintate conice,
BuletinulInstitutului de Petrol si Gaze, Ploiesti, nr.2/1981
[5] GRIGORE, N. a. o. Determinarea grafica aunghiului de
inclinare de divizare interior al danturii
rotilor dintate conice, Buletinul Institutului de Petrolsi Gaze,
Ploiesti, nr.1/1982
[6] GRIGORE, N. a. o. Calculul grafic al unghiurilorde inclinare
de divizare al danturilor conice
circulare, Studii si Cercetari de Mecanica Aplicata
nr.6/1982[7] GRIGORE, N., Organe de Masini, Transmisii
Mecanice, Editura Universitatii din Ploiesti, Ploiesti,2003, pp
229-278
[8] GRIGORE, N., a. o., Metoda si program pentrucalculul
parametrilor de reglaj ai masinii de danturat
conic in arc de cerc 528 SARATOV, VolumulLucrarilor Sesiunii
Stiintifice 45 ani de invatamant
superior la Galati 28-29 oct. 1993, Galati, 1993[9] RADULESCU,
Gh., a. o., Indrumar de proiectare in
constructia de masini, Editura Tehnica, Bucuresti,1986, pp
59-84
[10] * * *, Cartea masinii de danturat conic in arc de cerc
528 SARATOV
CORRESPONDENCE
Niculae GRIGORE, Prof. D.Sc. Eng.PETROLEUM-GAS University
ofPloiesti, Faculty of Mechanical andElectrical Engineering,
GeneralMechanics Department, Bucuresti Blvd,
no. 39, Ploiesti 100680,
[email protected]
Adrian CREITARU,Lecturer. D.Sc. Eng.PETROLEUM-GAS University
of
Ploiesti, Faculty of Mechanical and
Electrical Engineering, General MechanicsDepartment, Bucuresti
Blvd, no. 39,Ploiesti 100680, [email protected]
