Rack-Cutter Fillet Effect on Gear Undercutting

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Regarding the Effect of the Rack-Cutter Fillet on the Undercutting of Gears
Conference Paper · May 2012
DOI: 10.1007/978-94-007-6558-0_14
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Ognyan Alipiev
Sergey Antonov
University of Ruse Angel Kanchev
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Tanya Grozeva
University of Ruse Angel Kanchev
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POWER TRANSMISSIONS
The 4th International Conference on Power Transmissions, June 20-23, 2012,
Sinaia, ROMANIA
REGARDING THE EFFECT OF THE
RACK - CUTTER FILLET ON THE
UNDERCUTTING OF GEARS
Ognyan ALIPIEV 1, Sergey ANTONOV2, Tanya GROZEVA3
1, 2, 3
University of Ruse, BULGARIA
[email protected], [email protected], [email protected]
Abstract. In this work a generalized approach for defining the phenomenon
”undercutting of involute teeth” is proposed, where besides the traditional
boundary case, called “undercutting – type I”, additionally two more boundary cases, called “undercutting – type II” and “undercutting – type III”
are included. According to this approach the nontraditional cutting – type II
and type III is caused by the rack-cutter fillet, when the trajectories of some
points cross respectively: 1) the radial line of the gear (the line connecting
the gear centre and the starting point of the involute profile); 2) the involute
profile of the cut teeth. The parametric equations of the so called ”boundary
fillets” of the type II and type III of the rack-cutter are specified and in an
evident state the additional boundary condition for avoiding the undercutting of teeth is drawn up. The maximum value of the radius of the the
rack-cutter fillet at which the cut teeth are not undercut, is specified.
Keywords. Gear, Involute profile, Undercutting of teeth, Rack-cutter fillet,
1. INTRODUCTION
In the traditional theory of involute meshing (Litvin, 1968; Bolotovskii, 1986;
GOST 16532–70, 1970; Colbourne, 1987; Alipiev et al., 2011) the condition that
defines the nonundercutting of teeth is defined with the examination of the
meshing of the rectilinear profile of the rack-cutter with an involute profile of the
manufactured gear. In this case the undercutting of teeth, called by the authors
„undercutting – type I ”, is avoided when the trajectories of all points of the
rectilinear profile of the rack-cutter cross the line of action. When using the
traditional approach the influence of the rack-cutter fillet on the teeth undercutting
is not taken into account and the obtained results are correct only in cases when the
radius of the rack-cutter fillet is not too large. When this radius exceeds a specified
value (Alipiev, 2011) it might turn out that the traditional condition for
nonundercutting is satisfied but in reality the involute gear teeth are undercut. For
that reason, in the present work a new generalized method for defining the
undercutting of gears is proposed, where together with the traditional boundary
case (undercutting – type I ), two more boundary cases defined as „undercutting –
type II ” and „undercutting – type III ” are taken into consideration. In these two
additional cases the undercutting is done by the rack-cutter fillet and not by its rectilinear profile. Under this the undercutting – type II is characterized by the decrease of the tooth thickness at their bottom without cutting the involute profile, while
by undercutting – type III, a part of the initial area of the involute profile is cut.
2. UNDERCUTTING OF TEETH AT MESHING OF AN
INVOLUTE GEAR WITH THE RACK - CUTTER
2.1. Undercutting - type I (traditional case)
The undercutting of teeth – type I of the involute gear is obtained when at its
meshing with the rack-cutter (Fig. 1а), points of the rectilinear profile KE of the
2
Ognyan ALIPIEV, Sergey ANTONOV, Tanya GROZEVA
Fig. 1. Conditions of Nonundercutting – type I
a) X < Xmin; b) X > Xmin; c) X = Xmin
tooth cutter are situated under the line of action AB. Then the trajectories of these
points (lines parallel to the line n-n) do not cross the line of action AB (they cross
its extension), as a consequence of which the basic theorem of meshing is not
satisfied when realizing the contact between the pair profiles. In this case the
rectilinear area КЕ of the rack-cutter becomes a non-operating area that penetrates
into the bottom of the processed tooth and cuts a part of an involute profile.
In order to avoid the undercutting – type I it is necessary that the tip line g-g of the
rack-cutter cuts the line of action AB (Fig. 1b). Otherwise, when the crossing point
L lies outside the line of action on the extension of the line PA (Fig. 1а), the cut
teeth are undercut. This means that the condition for nonundercutting – type I
generally can be expressed by the inequality
(1)
PA ≥ PL ,
and taking into account the specified distances marked on Fig. 1, the traditional condition for undercutting - type I finally is written in the following way (Litvin, 1968)
x ≥ ha* − 0,5 z sin 2 α .
(2)
In the inequality (2) x = X / m is a shift (modification) coefficient of the rackcutter, ha* = ha / m – a coefficient of the height of the addendum, z – teeth number of
the gear, α – pressure (profile) angle of the rack-cutter, m – a module of the gear.
The smallest displacement X min of the rack-cutter, eliminating the undercutting –
type I, is defined by the equation
Regarding the effect of the rack-cutter fillet on…
X min = xmin m = ( ha* − 0,5 z sin 2 α ) m .
3
(3)
This displacement corresponds to the so called boundary case – type I (Fig. 1c),
where the tip line g-g passes through the boundary point A (the point where the
line of action contacts the base circle of a radius rb ).
2.2. Undercutting - type II and type III
As already mentioned, the undercutting – type I and type II are caused by the rackcutter fillet AF (Fig.2) in the process of teeth cutting. When this fillet is a circle of
a small radius ρ1 (Fig. 2a) the cut teeth are not undercut. Then the fillet fa of the
gear teeth does not cross the radial line, passed from the centre О to the starting
point a of the involute profile ае (at X=Xmin point a lies on the base circle of a
radius rb ).
Fig. 2. Types of undercutting of the rack-cutter fillet
a) nonundercutting profile; b) undercutting – type II c); undercutting – type III
At comparatively larger radius ρ2 of the fillet AF (Fig. 2b), an undercutting – type
II is obtained, where the fillet fa of the cut tooth crosses (cuts) the radial line Оа,
but does not cross the involute profile ае. This means that in the presence of
undercutting – type II the tooth thickness at the bottom decreases without cutting
an involute profile in the vicinity of its starting point a.
When the radius ρ3 (Fig. 2c) of the rack-cutter fillet increases considerably, the
fillet fq of the gear crosses the radial line Оа, as well as the involute profile. In this
case besides the decrease of the tooth thickness at the bottom, the area aq of the
involute profile ае is also cut.
The essence of the undercutting – type II and type III is explained on Fig. 3, where
the undercutting – type I is avoided by a positive displacement of the rack-cutter at
a distance Xmin. At this boundary displacement, the tip line g-g of the rack-cutter
crosses the line of action AB in its starting point A.
In order to define the maximum radius of the rack-cutter fillet АF, corresponding to
the boundary case where there is no undercutting – type II, on Fig. 3a additionally
is drawn the curve qII, called a boundary fillet – type II. It is obtained as an envelope of the relative positions that take the radial line l (the line aO) of the gear in the
plane of the rack-cutter, when realizing the meshing between the rectilinear profile
АЕ of the rack of the involute profile ае of the gear. In other words the profiles l
and qII are also conjugated profiles at rolling without sliding of the centroid line
n-n of the а rack-cutter on the reference circle of the gear of a radius r. Knowing
4
Ognyan ALIPIEV, Sergey ANTONOV, Tanya GROZEVA
the curve qII allows us to define the following boundary condition: the undercutting
– type II is avoided if the real rack-cutter fillet АF is placed internally regarding
the boundary fillet qII (in the material of the cutter). On Fig. 3a the curve АF is
placed externally regarding the curve qII, as a result of which gear teeth are
undercut – type II.
Analogously the condition for nonundercutting – type III is defined by drawing a
curve qIII (Fig. 3b), called a boundary fillet – type III. In this case the curve qIII is
obtained as a trajectory (drawn in the plane of the rack-cutter) of the point а from
the plane of the reference circle of a radius r, rolling without sliding on a reference
circle on the line n-n. As point а lies on the internal side of the reference circle, the
Fig. 3. Undercutting of teeth by rack-cutter fillet
a) type II ; b) type III
drawn trajectory represents a shortened epicycloid. The same curve qIII, connected
without moving with the rack-cutter, can be considered also as a conjugated curve
of the starting point а of the involute profile. This means that if the real rack-cutter
fillet coincides with qIII, at each moment it will contact with point А and will not
cut the involute profile aq. Therefore the undercutting – type III is avoided if the
real rack-cutter fillet АF is placed internally regarding the boundary fillet qIII. In
the case shown on Fig. 3b this condition is not satisfied and as a result the gear
teeth are undercut – type III .
2.3. Еquations of boundary fillet curves
Boundary fillet – type II . The equations of this curve are obtained using the
theory of plane meshing [Litvin, 1968; Litvin, 2004], where one of the two meshed
Regarding the effect of the rack-cutter fillet on…
5
profiles appears and the other one is obtained as an envelope of the relative
positions which the specified profile occupies in the plane of the searched profile.
In this case (Fig. 4) the specified profile is the radial line l of the gear and the
searched profile is the boundary fillet qII of the rack-cutter.
In order to solve the problem of the geometrical synthesis two mobile coordinate
systems are introduced: XlOlYl – connected with the gear (the specified profile l);
XII OYII – connected with the rack-cutter (the searched profile qII). As the axis OXII
Fig. 4. A boundary fillet of the rack-cutter – type II
(the centroid line) of the rack-cutter rolls without sliding on the reference circle (of
a radius r) of the gear, the displacement s of XII OYII is synchronized with the
rotation of XlOlYl at an angle φ, where s = r φ. The place of the contact points of
profiles l and qII in the motionless plane, designed with K, A, etc. is defined as from
the pitch point P perpendicular lines to the respective positions which the radial
line l takes, are dropped. In fact the equations of the curve qII are derived by
defining the place of the same contact points in the rectilinear moving coordinate
system XII OYII.
After executing the respective transformations and conversions [Litvin, 1968], the
parametric equations of the boundary fillet – type II are finally written as follows:
X II = r (ϕ − cos ϕ sin ϕ ) = X II (ϕ ) ,
(4)
YII = − r sin 2 ϕ = YII (ϕ ) ,
where φ is the angular parameter of the curve, and r – the radius of the reference
circle of the gear, calculated by the equation
r =mz 2.
(5)
The obtained curve qII is divided by point А to two areas: AN and AM. On Fig. 4 it
is seen that only the area AM appears as the real boundary rack-cutter fillet. This
means that when drawing the real curve qII , the parameter φ gets an initial value of
φ = α (point A) and increases in the direction from point А to point M.
From the differential geometry it is known that the radius of a curvature ρ on each
curve, specified as X=X(φ), Y=Y(φ), is defined from the equation
( X& 2 + Y& 2 ) 3 2
ρ = & && && & = ρ (ϕ ) ,
(6)
( XY − XY )
where X& , Y&, X&&, Y&& are the first and second derivatives to the parameter φ.
6
Ognyan ALIPIEV, Sergey ANTONOV, Tanya GROZEVA
Taking into account that
X& II = 2r sin 2 ϕ , X&& II = 4r sin ϕ cosϕ , Y&II = −2r sin ϕ cosϕ , Y&&II = −2r cos 2ϕ
(7)
for the equation of the radius of the curvature of the curve qII , it is finally obtained
ρ II = 2 r sin ϕ = m z sin ϕ .
(8)
Boundary fillet – type III. It is obtained as a trajectory of the point A of the
coordinate system XAOAYA connected with the gear (Fig. 5), drawn in a coordinate
Fig. 5. A boundary fillet of the rack-cutter – type III
system XIII OYIII , connected with the rack-cutter. The obtained trajectory qIII , as it
was already explained, represents a shortened cycloid, whose parametric equations
can be written as follows
X III = rϕ − rb sin ϕ = X III (ϕ ) ,
(9)
YIII = − r + rb cosϕ = YIII (ϕ ) ,
where rb is the radius of the base circle, defined by the formula
rb = r cosα = 0,5m z cosα .
(10)
In the initial position of the coordinate systems, when at φ = 0, point А coincides
with point А'', and at φ = α – with point А'. In that case, point А' appears as an
inflection point, which divides the curve qIII in two parts: – a concave area NА' and
a convex area А'M. In this case it should be taken into account that only area А'M is
the real rack-cutter fillet – type III .
The equation of the radius of the curvature of the curve qIII is obtained in analogous
way as the curve qII , with the use of the equation (6). In this case the first and
second derivatives of XIII and YIII to φ are defined from equations
X& III = r − rb cosϕ , X&& III = rb sin ϕ , Y&III = − rb sin ϕ , Y&&III = − rb cosϕ ,
(11)
and the radius of the curvature ρIII = ρIII (φ) of the curve qIII is obtained by the
formula
(r 2 − 2 r rb cos ϕ + rb2)3 2 m z(1 − 2 cos α cos ϕ + cos2 α)3 2
·
=
ρIII =
(12)
2 cos α(cos α − cos ϕ)
rb2 − r rb cos ϕ
2.4. Conditions of Nonundercutting - type II
In order to clarify the causes by reason of which the rack-cutter fillet undercuts the
gear teeth, on Fig. 6 both boundary curves qII and qIII are drawn simultaneously in
Regarding the effect of the rack-cutter fillet on…
7
the current position where their common contact point coincides with the starting
point А of the line of action АВ (the position where φ = α ). In this position the
radial line ОАЕ (Fig. 6а), representing simultaneously a rectilinear profile of the
rack-cutter, crosses the curve qIII in its inflection point, where in this case coincides
with point А and appears as a contact point of ОАЕ with the curve qII .
Fig. 6. Radius of the rack-cutter fillet
From Fig. 6b it becomes clear that if the rack-cutter fillet is an arc of a circle of a
small radius ρ1 there exists no undercutting – type II and type III, because in this
case the arc AF1 lies on the internal side of curves qII and qIII . When the rack-cutter
fillet is positioned between both curves qII and qIII (the arc AF2 of a radius ρ2>ρ1) an
undercutting – type II appears, and when the rack-cutter fillet is placed between the
curve qIII and the line ОЕ (the arc AF3 of a radius ρ3>ρ2) besides an undercutting –
type II, an undercutting – type III is derived.
In the case where the rack-cutter fillet is profiled on an arc from a circle, the
following boundary condition is defined: the undercutting - type II is avoided if the
radius of the fillet is smaller or equal to the radius of the curve ρII,A of the boundary
fillet qII in point А. Since in point А of the curve qII the value of the angular
parameter is φ = α, for the radius of the curve ρII,A in this point according to the
equation (8), it is obtained
ρ II, A = mz sin α .
(13)
Then the condition for the nonundercutting – type II , defined by the inequality
ρ ≤ ρII,A , is written finally as follows
ρ * ≤ z sin α ,
(14)
where ρ*= ρ / m is a coefficient of the radius of the circle, on which the rack-cutter
fillet is profiled. The condition (14) shows that the undercutting – type II depends
only on the teeth number z of the gear and the profile angle α of the rack-cutter.
In Table 1 the maximum values of the dimensionless coefficient ρ*max, are shown,
corresponding to profile angles used in practice for different number of cut teeth.
In order to avoid the undercutting – type II it is enough for the radius ρ, on which
the rack-cutter fillet is profiled, to be smaller or equal to the respective value ρ*max ,
multiplied by the module m of the gear (ρ ≤ ρ*max m). From Table 1 it is seen that
ρ*max increases when the pressure angle α and teeth number z are increased. Besides,
from the specified values it becomes clear that if a standard rack-cutter is used, for
which α=20° and ρ* = 0.38, the cut teeth are not undercut – type II and type III (at
z = 5 and α=20° → ρ* max = 1.71).
Ognyan ALIPIEV, Sergey ANTONOV, Tanya GROZEVA
8
* = z sin α
Table 1. Maximum coefficient fillet radius of the rack-cutter – ρ max
Number
of teeth
z
5
10
20
40
80
160
14°30'
1.252
2.504
5.008
10.015
20.030
40.061
Profile angle of the rack-cuter - α
15°
17°30'
20°
22°30'
25°
1.294
1.504
1.710
1.913
2.113
2.588
3.007
3.420
3.827
4.226
5.176
6.014
6.840
7.654
8.452
10.353 12.028 13.681 15.307 16.905
20.706 24.056 27.362 30.615 33.809
41.411 48.113 54.723 61.229 67.619
28°
30°
2.347 2.5
4.695 5.0
9.389 10.0
18.779 20.0
37.558 40.0
75.115 80.0
Here it is important to note that if the cut teeth are not undercut – type II, they are
also not undercut – type III. Therefore the satisfying of the boundary condition (14)
guarantees the nonundercutting of teeth of type II, as well as of type III.
3. CONCLUSION
The carried out investigations connected to the undercutting of teeth of involute
gears when cut by the rack-cutter, result in the following conclusions:
1. The undercutting of teeth is done from the rectilinear profile of the rack-cutter
(traditional case – type I), as well as by the fillet, on which are cut the teeth of its
cutting teeth crests (the non-traditional case) are cut.
2. The undercutting of teeth, caused by the rack-cutter fillet is found in two
variants, defined in the present work as „undercutting – type II” and „undercutting – type III ”.
3. In the presence of the undercutting – type II the teeth thickness in their bottom
is decreased without cutting their involute profile, and in the presence of
undercutting – type III a part of the involute profile is additionally cut.
4. The condition for nonundercutting – type II is defined uniquely by two independent parameters: the number z of the cut teeth and the profile angle α of the rackcutter.
5. In order to avoid the undercutting of the involute teeth, it is necessary to satisfy
the traditional condition (2) as well as the boundary condition (14).
References
ALIPIEV, O. L. (2011). Geometric design of involute spur gear drives with symmetric and asymmetric teeth using the Realized Potential Method, In:
Mechanism and Machine Theory, Volume 46, Issue 1, pp.10-32.
ALIPIEV, O., ANTONOV, S., GROZEVA, T. (2008). Determining the Position of
Tooth Undercut Border Point in Involute Gears with Spur Teeth, Shaped by
Rack-Cutter. In: Mechanics of Machines, Vol. 75, pp. 56-62, (in Bulgarian).
BOLOTOVSKII, I. A. et al. (1986) Reference Book in Geometric Calculation of
Involute and Worm Gearings, Mashinostroyenie, Moscow, (in Russian).
COLBOURNE, J. R. (1987). The Geometry of Involute Gears, Springer–Verlag,
New York.
GOST 16532–70 (1970). Cylindrical Involute External Gear Pairs – Calculation of
Geometry, Moscow, Russia.
LITVIN, F. L. (1968). Theory of Gearing, Ed. Nauka, Moscow, (in Russian).
LITVIN, F., FUENTES, A. (2004). Gear Geometry and Applied Theory, Cambridge University Press, Cambridge.
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