A method of generating crowned pinion tooth surfaces using a surface of revolution is developed. The crowned pinion meshes with a regular involute gear and has a prescribed parabolic type of transmission errors when the gears operate in the aligned mode. When the gears are misaligned the transmission error remains parabolic with the maximum level still remaining very small (less than 0.34 arc sec for the numerical examples). Tooth contact analysis (TCA) is used to simulate the conditions of meshing, determine the transmission error, and determine the bearing contact

Handschuh, R. F.

Litvin, F. L.

Zhang, J.

English

NASA Technical Reports Server

tNASA
Technical Memorandum 100260
AVSCOM
Technical Report 87-C-34
Generation of a Crowned Pinion Tooth
Surface by a Surface of Revolution
F.L. Litvin and J. Zhang
University of Illinois at Chicago
Chicago, Illinois
and
R.F. Handschuh
Propulsion Directorate
U.S. Army Aviation Research and Technology Activity--AVSCOM
Lewis Research Center
Cleveland, Ohio
(_ASA-T/'I-1CC266) 6EN_aICN C_ a C_OWNJ_D
_tV6L[_ ItS |_A_) 15 r CSCL 13I
G3/._ 1
N89-1C282
Unclas
0168503
September 1988
AVIATION
SYSTEMS COMMAND
AVIATION F_T ACTIVITY
https://ntrs.nasa.gov/search.jsp?R=19890000911 2020-03-20T06:01:04+00:00Z

GENERATION OF A CROWNED PINION TOOTH SURFACE BY A SURFACE OF REVOLUTION
F.L. Litvin and J. Zhang
University of Illinois at Chicago
Chicago, Illinois 60680
and
R.F. Handschuh
Propulsion Directorate
U.S. Army Aviation Research and Technology Activity - AVSCOM
Lewis Research Center
Cleveland, Ohio 44135
O0
CO
CO
c,3
I
i,i
SUMMARY
A method of generating crowned pinion tooth surfaces using a surface of
revolution is developed. The crowned pinion meshes with a regular involute
gear and has a prescribed parabolic type of transmission errors when the gears
operate in the aligned mode. Nhen the gears are misaligned the transmission
error remains parabolic with the maximum level still remaining very small (less
than 0.34 arc sec for the numerical examples). Tooth contact analysis (TCA) is
used to simulate the conditions of meshing, determine the transmission error,
and determine the bearing contact.
INTRODUCTION
Parallel axis gearing is very sensitive to misallgnment of the gear rota-
tion axes. Misalignment can cause edge loading of the teeth and thus premature
failure of the gear components. Crowning one or both of the gear pairs can be
used. The surface can be modified in the longitudinal direction or in the pro-
file direction, or a combination of longitudinal and profile modification can
be used to modify the surface. All modifications are directed at finding a
satisfactory amount of crowning for a given degree of misalignment.
Another condition, kinematic error (transmission error), results when
gears are crowned and their axes operate in a misaligned mode. This condition
can increase the gear mesh generated noise. The kinematic error that can
result can be severe enough to cause an interruption or discontinuity during
the meshing process. The discontinuity causes impacting of the teeth instead
of the relatively smooth action that is typically found in parallel axis gear-
ing operating under light load conditions.
A method for crowned gear generation is presented. The method only
requires that the pinion be crowned. The pinion tooth surface is modified in
the profile and longitudinal directions. The gear that is in mesh with the
pinion is a regular involute.
The pinion crowning method used is as follows"
(I) The pinion generating tool is provided with a surface of revolution
that is slightly modified from a cone surface.
(2) The pinion tooth surface and the tool generating surface are in line
contact at every instant.
(3) The relationship between the motions of the tool and the pinion in the
process of generation is the sameas that between the rack cutter and the gen-
erated spur gear.
(4) The surface of revolution of the generating tool is rotated about its
axis only to provide the desired velocity of cutting.
The synthesis of spur gears generated by this method is based on determining
the appropriate principal curvatures of the generating surface.
GENERATIONMETHOD
The generation method to be discussed has the following requirements:
(I) obtain a localized bearing contact and (2) reduce the gear sensitivity to
misalignment. The generation method shown in figure l (coned surface for gen-
erating pinion, rack cutter for generating gear) satisfies only the first
requirement - localization of the bearing contact. However, it is preferable
to start discussions with this generation method and then to extend the analy-
sis to a more sophisticated generation method to satisfy both requirements.
The basic idea of the first generation method is the application of two
rigidly connected generating surfaces - a plane and a cone. The plane is the
surface of a regular rack cutter. Both generating surfaces are in tangency
along a straight line - the generatrix of the cone surface. The cone generates
the pinion tooth surface, and the plane generates the gear tooth surface. In
the process of generation the rack cutter and the cone perform a translational
motion while the pinion and the gear rotate about their axes (fig. 2). The
rotation of the cone about its axis c-c is not related to other motions that
must be provided to generate the tooth surface. The angular velocity of the
cone depends only on the desired velocity of cutting. The tool for crowning
the pinion can be designed as a grinding wheel or as a shaver. The pinion isgenerated one side at a time.
The described process of pinion crowning by a cone provides an involute
shape of the pinion tooth surface in its middle section. The crowned pinion
and the involute gear, if they are not misaligned, can transform rotation with-
out transmission errors and their bearing contact can be localized. However,
these gears whenmisaligned wi11 transform rotation with transmission errors of
type 1 (fig. 3(a)). To avoid the discontinuity during meshing, a surface of
revolution must be used instead of a cone surface. This surface is slightly
modified from a regular cone surface, and its application for crowning provides
kinematical errors of type 2 (fig. 3(b)). The generated pinion tooth shape in
the middle cross section is no longer an involute curve. The pinion and the
gear, even if they are not misaligned, transform rotation with a low level of
transmission errors that resemble a parabolic function. If the gears are mis-
aligned the transmission errors are still a parabolic function.
COORDINATESYSTEMSANDTOOLSURFACES
While considering the generation of gear tooth surfaces, we use coordinate
systems SI, S2, Sc, and Sf (fig. 2) where S1 and S2 are rigidly connected
to the pinion and gear being generated; Sc is rigidly connected to the cutters
that perform their translational motion as one rigid body; and Sf is the
fixed coordinate system. The tool displacement s and the angles of gear
rotation are related as follows:
s = ri¢ P : r25G
where r I and r 2 are the radii of the pinion and gear pitch circles,
respectively.
While considering the meshing of the generated gears, we will use the same
coordinate systems SI, S2, and Sf (fig. 4). In order to put the generated
surfaces (fig. 2) in contact, the direction of rotation of the pinion and gear
are reversed (fig. 4). Angles Sp and SG are the angles of pinion and gear
rotation while they are generated by the tools (fig. 2); $I and $2 are the
angles of pinion and gear rotation while the gears are in direct meshing with
each other. Angles $G and Sp are related by a linear function, but the
function $2($i ) is a nonlinear one. The deviation of function $2($i ) from
the linear one depends on the deviation of the tool surface of revolution from
a regular cone surface.
Figure 5 shows three different cases of generating tool surfaces. In
case 1 (fig. 5(a)), both rigidly connected generating surfaces _p and [G
are planar (rack cutter) and coincident with each other. The generated gears
will be regular involute surfaces. Thus, the pinion is not crowned, and the
bearing contact is not localized.
In case 2 (fig. 5(b)), Sp is a regular cone surface and ZG is a plane.
Surfaces [p and EG are in contact at the generatrix of the cone. The tool
surface Sp generates the crowned pinion tooth surface, and ZG generates the
regular involute gear tooth surface. The bearing contact of the pinion and
gear tooth surfaces is localized, but a predesigned deviation of function
$2($i ) from the linear function
0 N1
@2= _22$1 (l)
cannot be obtained where N1 and N2 are the tooth numbersof the pinion and
gear, respectively.
In case 3 (fig. 5(c)), the pinion generating surface Sp is a surface of
revolution, and the gear generating surface EG is again a plane. The tool
surfaces are in tangency at a point instead of at a line as in case 2. The
pinion is provided with a crowned tooth surface. The bearing contact of the
gear tooth surfaces is localized, and function $2($i ) can deviate from the
linear function (eq. (I)) by the appropriate value of the radius of the arc of
the surface of revolution. This arc is the axial section of the surface of
revolution.
LOCALSYNTHESIS
The gear local synthesis must provide the following conditions of meshing
at the mean contact point P (fig. 4) and its neighborhood"
(I) The gear tooth surfaces are in contact at P.
(2) The instantaneous gear ratio is not constant but is represented by a
function m2l(¢l) where
d
m21(¢ I) - [¢2(¢ )]
d¢ 1 l
(2)
(3) The instantaneous value of m21(¢ I) at
prescribed value
Nl
m21(O) -
N2
¢] = 0 must be equal to the
(3)
(4) The function ¢2(¢i) is represented as
N1
¢2(¢I) = _22 ¢I + &¢2(¢I ) (4)
where &¢2(@i ) is the predesigned function. Assuming that
bolic function glven as
&¢2(¢ I) : -a¢_
_¢2(¢i) is a para-
(5)
we obtain, when ¢I = O,
d Nl
m21(O) = dT l [¢2(¢I )] - N2
(6)
and
dm21 (0) d2
d¢l - m'_l(O) - d¢_ [@2(¢i )] : -2a
(7)
This means that a predesigned parabolic function for transmission errors satis-
fies the requirement that the instantaneous gear ratio m21(O) is equal to the
I
given value (eq. (6)) and provldes a negative value for the derivative m21(O).
(5) When the generatlng surface of revolution Is used, the existence of
the parabolic function for transmission errors can only be guaranteed in the
neighborhood of the mean contact polnt. The magnitude of m21(O) can be con-
trolled by choosing the appropriate curvature radlus p of the axial section
of the surface of revolution. The determination of p, which is based on the
equation given in references l and 2, relates the principal curvatures and
directions and m_l(O). For the case of crowned spur gears, this equation is
represented as
4
fan,c(N )1 11 + rlsin 9c (8)
where _c is the pressure angle and p the radius of the axial section arc of
generating surface (fig. 7).
CRONNED PINION TOOTH SURFACE EQUATIONS
Consider that the tool surface (a surface of revolution) is represented in
an auxiliary coordinate system by the vector function £a(ep,6) where _p and
are the surface coordinates. Matrix [Mca] describes the installment of the
tool system Sa in coordinate system Sc. Using the coordinate transformation
Sa to Sc given by
[r c] = [Mca] [r a]
we can represent the tool surface in Sc by the vector function Za(ep,B).
While the coordinate system Sc translates, the pinion being generated
rotates (fig. 2). The installment of the tool with respect to the pinion is
shown in figure 6. Equations of the generated pinion are represented as
[r 11 = [Mlc] [r c]
(9)
(10)
f(¢p,ep,B) = N(c) • v(cP) = 0 (II)
Equation (10) describes coordinate transformation in transition from Sc to
SI. Equation (II) is the equation of meshing, N(C) is the tool surface
normal, and v(cP) is the relative velocity of the tool with respect to the
pinion.
Equations (I0) and (11) represent the pinion tooth surface in the three-
parameter form with the additional relation between the parameters. Thus,
Z I : Zl(ep,_,¢ P)
f(Qp,Bp,_) = 0
(12)
By eliminating one of these parameters, we are able to represent the pinion
tooth surface in the traditional two-parameter form. More details of pinion
surface generation are contained in the appendix.
TCA PROGRAM
TCA programs have been developed based on the principles presented in ref-
erence 3. These programs simulate the conditions of meshing and determine the
transmission errors caused by gear misalignment.
5
A numerical example is given for two types of misalignment. The follow-
ing conditions are given: !_lon and gear tooth numbers, N1 : 20, N2 = 40;diametral pitch, Pn : I0 in ; pressure angle, _c = 20°; half apex angle of
auxiliary cone, m = 80°; height of auxil|ary cone, d : 0.176 in.; radius of
generatrix arc, p : 500 in. (figs. 7 and 8). The developed TCAprogram has
been applied to evaluate the kinematic error for the following:
(I) The change of the center distance is &C/C= l percent, where
C = (N1 + N2)/2Pn. The gear axes are not parallel but they cross, and the
twist angle is 5 to I0 arc min. The kinematic error is of type 2 (fig. 3(b)).
The maximumvalues of kinematic error for these cases are 0.30 and 0.31 arc
sec, respectively.
(2) The changeof center distance is &C/C= 1 percent, where
C = (Nl + N2)/2Pn. The gear axes are not parallel but they intersect, and the
intersecting angle is 5 and lO arc min. The kinematic error is also type 2(fig. 3(b)). Corresponding maximumvalues of the kinematic error are 0.34 and
0.32 arc sec, respectively.
The results of this numerical example i11ustrate that the proposed method
for generating a crowned plnlon surface can avoid the appearance of a lead
function of transmission errors and can reduce the level of these errors.
CONCLUSIONS
A method for generating a crowned pinion tooth surface using a surface of
revolution has been developed. The meshing gear is generated by using a rack
cutter. The following are the conclusions of this study:
I. A surface of revolution is used to generate a crowned pinion surface.
The pinion meshing with a regular involute gear provides a prescribed parabolic
function of transmission errors whenthere is no misa]ignment.
2. Whenthe modified gears are misaligned, the transmission errors remain
a parabollc function without discontinuities.
3. The maximumlevel of kinematic error, due to misalignment, remains low.
APPENDIX- EQUATIONSOFPINIONGENERATINGSURFACEANDPINIONTOOTHSURFACE
As shown in figure 7(a), arc B and straight line C are the genera-
trices of the revolute surface and the cone. Arc B and its normal can be
represented in coordinate system Se as follows"
[r e] :
"p[cos (_ + B) - cos _] + aI sin ="
p[sin (_ + 6) - sin _] + a2 cos
0
(13a)
[r e] =
a I sin = - 2p sin (BI2) sin (= + _12)
a2 cos _ + 2p sin (BI2) cos (_ + 612) (13b)
cos (_ + B)1[ne] = sin (_ + B)
0
(14)
The surface of revolution, which is generated by rotating the circular arc
B about the Ye-aXis, can be represented in coordinate system Sa as follows
(see fig. 7(b))"
[r a] : [Mae] [re]
Ina] = [Lae] [ne]
(15)
where
[M ] =
ae
cos ep o sin ep O
0 1 0 0
-sin ep o cos ep o
o o o 1
cos ep 0 sin Op]
[Lae] = 0 1 0
-sin Op 0 cos ep
Equations (13) to (15) yield
x a
[ra] : Ya :
za
.I
[a I sin _ - 2p sin (BI2) sln (_ + BI2)] cos Op
a2 cos _ + 2p sin (_12) cos (_ + BI2)
[a I sin _ - 2p sin (BI2) sin (_ + _12)] sin Op
(16)
'cos (_ + B)[n a] = sin (_ + _)
LCOS (= + _)
cos epl
sin ep
trated in figure 8.
by equation (9) as
where
[Mca] :
Also, we have
Installing the tool revolute surface in coordinate system
The coordinate transformation from Sa to
where
[rc] : [Mca] [ra]
cos (_ - 9c ) sin (_ - _c ) 0
-sin (_ - _c ) cos (_ - _c ) 0
0 0 1
0 0 0
Inc] = [Lca] [na]
cos (_ - _c )
[Lca] : l -sin (_0 _c)
Sc is illus-
Sc is given
(17)
-a I sin _c - d sin (_ - _c )
a 1 cos _c - d cos (_ - _c )
(18)
cos (o_ - _c )
0
Consldering equations (ll), (17), and (18) and the pinion generating proc-
ess shown in figure 2, we can get the followlng equation of meshing:
f(B,ep,¢p) = p sin B cos ep - aI sin _ sln (_ + B)(cos ep - I)
+ r1¢p[sin (_ + B) cos (_ - _c ) - cos ep cos (_ + B) sin (_ - _c )] = 0 (19)
8
The pinion tooth surface can be represented in coordinate system
[ri] = [Mic] [rc] = [Mlf] [Mfc] [rc]
where the transformation matrices [Mfc] and [Mlf] are given by
.]
[Mfc] : 0
0
.0
0 0 -rl¢ p"
l 0 r1
0 1 0
0 0 1
Sl as
(20)
(21)
icos,ps°i][Mlf ] : -sin Cp cos ¢p 00 0 1
0 0 0
(22)
REFERENCES
I. Litvin, F.L." Theory of Gearing. 2nd ed., Moscow, Nauka, 1968 (in
Russia).
2. Litvin, F.L.; and Zhang, J." Spur Gears: Optimal Geometry, Methods for
Generation and Tooth Contact Analysis (TCA) Program. NASA CR-4135, 1988.
FIGURE 1. - GENERATING SURFACES FOR P[NION (CONE) AND GEAR (RACK).
Yc
O(
PINION _
Yf /-CONE CUTTER //- v = glrl = g2r 2
/
x 1 / /
xC
m,
xf
/
L_ RACK CUTTER
FIGURE 2. - GEOMETRY RELATIONSHIP BETWEEN GENERATING (CUTTERS) AND GENERATED (PINION AND GEAR) SURFACES.
I0
A<p
®I / /\
/
(a) DISCONTINUOUSTRANSMISSIONERROR (TYPE 1). (b) PARABOLICTRANSMISSIONERROR(TYPE 2).
FIGURE 3. - TRANSNISSIONERRORTS(_2) AS FUNCTION OF PINION ROTATION ((I)1),
Y2 Yf
/
Y Yl _
Of ,01 i
x 2
zf ,z 1
FIGURE 4. - COORDINATESYSTERSDESCRIBING RESHING OF PINION
AND 6EAR.
xf
x1
11
_P :_'G
(a) GENERATING SURFACE FOR INVOLUTE PINION AND GEAR (RACK CUTTERS).
(b) GENERATING SURFACE FOR PINION IS A CONE; GENERATING SURFACE FOR
GFAR IS A RACK.
_p
LGJ
(c) GENERATING SURFACE FOR PINION IS A SURFACE OF REVOLUTION;
GENERATING SURFACE FOR GEAR IS A RACK.
FIGURE 5. - THREE DIFFERENT SETS OF PINION AND GEAR GENERAIING
SURFACES.
\
\
FIGURE 6. - PINION GENERAIION TOOL AND PINION DURING SURFACE
CROWNING.
12
Ye ,Ya
Za Ze
FIGURE 7. - COORDINATE SYSTEM DESCRIBING SURFACE OF REVOLUTION
AND RACK CUTTER.
Rc I
b
xa
FIGURE 8,- COORDINATE SYSTEM DESCRIBING TRANSFORMATION FRO_ISYSTEM
Sa TO Sc.
13
Report Documentation PageNalional Aeronaulics and
Space Administration
1. Report No. NASA TM-100260 2. Government Accession No. 3. Recipient's Catalog No.
AVSCOM TR-87-C-34
4. Title and Subtitle
Generation of a Crowned Pinion Tooth Surface by a Surface of Revolution
7. Author(s)
F.L. Litvin, J. Zhang, and R.F. Handschuh
9. Performing Organization Name and Address
NASA Lewis Research Center
Cleveland, Ohio 44135-3191
and
Propulsion Directorate
U.S. Army Aviation Research and Technology Activity--AVSCOM
Cleveland, Ohio 44135-3127
t2. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, D.C. 20546-0001
and
U.S. Army Aviation Systems Command
St. Louis, Mo. 63120-1798
5. Report Date
September 1988
6. Performing Organization Code
8. Performing Organization Report No.
E-3888
10. Work Unit No.
505-63-51
11. Contract or Grant No.
13. Type of Report and Period Covered
Technical Memorandum
14. Sponsoring Agency Code
15. Supplementary Notes
F.L. Litvin and J. Zhang, University of Illinois at Chicago, Chicago, Illinois 60680 (work funded under NASA
Grant NAG3-655); R.F. Handschuh, Propulsion Directorate, U.S. Army Aviation Research and Technology
Activity--AVSCOM.
16. Abstract
A method of generating crowned pinion tooth surfaces using a surface of revolution is developed. The crowned
pinion meshes with a regular involute gear and has a prescribed parabolic type of transmission errors when the
gears operate in the aligned mode. When the gears are misaligned the transmission error remains parabolic with
the maximum level still remaining very small (less than 0.34 arc sec for the numerical examples). Tooth contact
analysis (TCA) is used to simulate the conditions of meshing, determine the transmission error, and determine the
bearing contact.
17. Key Words (Suggested by Author(s))
Gears
Gear tooth geometry
18. Distribution Statement
Unclassified- Unlimited
Subject Category 37
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Generation of a crowned pinion tooth surface by a surface of revolution

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