The design of spiral bevel and hypoid gears that have a shaft extended from both sides of the cone apex (straddle design) is considered. A main difficulty of such a design is determining the length and diameter of the shaft that might be undercut by the head cutter during gear tooth generation. A method that determines the free space available for the gear shaft is proposed. The approach avoids collision between the shaft being designed and the head cutter during tooth generation. The approach is illustrated with a numerical example

Bossler, Robert

Handschuh, Robert F.

Kieffer, Jonathan

Kuan, Chihping

Litvin, Faydor L.

English

NASA Technical Reports Server

NASA AVSCOM
Technical Mern6randuin 103621 _. Technical Report 90-C.010
Straddle Design of Spiral Bevel and
Hypoid Pinions and Gears
Faydor L. Litvin, Chihping Kuan, and Jonathan Kieffer ........
University of Illinois at Chicago
Chicago, Illinois
Robert Bossier ...........
Lucas Western Inc .....
City of Industry, California ........
and .........
Robert F. Handschuh_
Propulsion Directorate
• U.S. Army Aviation Systems Command
NASA _wis Research Center
Cleveland, Ohio ..... '_ _ _
2
SYSTEMS COMMAND
AVIATION R_,T ACTIVITY
_(NAgA-TH-IO362i) STRADDLe D_SIGN OF _PI_,t_L N91-14602
BEV£L AND HYPOID PINIONS ANn GEARS _
(Illinois Univ.) 12 p CSCL 13H _._eUnc I _ s
0317952G31__7
https://ntrs.nasa.gov/search.jsp?R=19910005289 2020-03-19T19:28:45+00:00Z

STRADDLE DESIGN OF SPIRAL BEVEL AND HYPOID PINIONS AND GEARS
Faydor L. Litvin, Chihping Kuan, and Jonathan Kieffer
University of Illinois at Chicago
Chicago, Illinois 60680
Robert Bossler
Lucas Western Inc.
City of Industry, California
and
Robert F. Handschuh
Propulsion Directorate
U.S. Army Aviation Systems Command
NASA Lewis Research Center
Cleveland, Ohio 44135-3191
I.D
!
Id.i
SUMMARY ------
The design of spiral bevel and hypoid gears that have a shaft extended
from both sides of the cone apex (straddle design) is considered. A main dif-
ficulty of such a design is determining the length and diameter of the shaft
that might be undercut by the head cutter during gear tooth generation. A
method that determines the free space available for the gear shaft is proposed.
The approach avoids collision between the shaft being designed and the head
cutter during tooth generation. The approach is illustrated with a numerical
example.
INTRODUCTION
The term "straddle design" means that the gear will be provided with a
shaft that extends from both sides of the toothed face of the gear for bearing
support. An example of a spiral bevel gear with such a shaft is shown in fig-
ure 1. The straddle design is preferred for heavily loaded gear trains, since
it enables the load to be split between shaft-mounted bearings on both sides of
the gear face. The straddle design requires the gear and the shaft to be made
as one piece. A design obstacle is the interference (collision) of the shaft
with the head cutter during tooth generation.
The valuable experience of Boeing Vertol Company in straddle design has
been described by R.J. Drago (ref. 1), who proposed a graphical method for
determining the dimensions of an integral shaft for spiral bevel gears.
In this report a method is proposed for detecting collision of the head
cutter with the shaft being designed. The procedure determines the optimal
length u of the shaft and its radius r (fig. 1). The solution is based on
determining a space in the plane of parameters r and u that is free from
interference with the head cutter. A computer program has been developed that
uses the basic machine tool settings for gear generation as input. The com-
puter program results are the diameter r and length u for the lines that
limit the space free from undercutting.
BASICMACHINETOOLSETTINGS
In order to begin to solve this problem, the kinematics of the tooth gen-
eration process must be modeled. The basic ideas from which this analytical
procedure is developed can be found in reference 2. The process of tooth gen-
eration will now be described. The following coordinate systems will be
considered:
(1) The coordinate system So(Xo,Yo,Zo), which is rigidly connected to
the cutting machine (figs. 2 and 3)
(2) The coordinate system Sc, which is rigidly connected to the cradle
that during tooth generation rotates about the Zo-axis with angular
velocity o(c)
(3) The auxiliary coordinate system SD, which is rigidly connected to
SO and the g_ar being generated and-rotates about Xp with angular
velocity _(P)
(4) The coordinate system S t , which is rigidly connected to the head cut-
ter (fig. 4) (note that the head cutter is mounted on the cradle and
rotates with it about the Zo-axis (fig. 3); in addition, the head cut-
ter rotates about the Zt-axis, but this rotation only provides the
required velocity for cutting and is not related to tooth generation)
(5) An additional coordinate system Sb (fig. 2), which is rigidly con-
nected to the cradle coordinate system Sc and is used to describe the
tilt of the head cutter with respect to the cradle
Coordinate systems S t and Sb initially coincide with each other. Their
orientation with respect to Sc is determined with the swivel angle j
(fig. 2). The tilt of the head cutter is accomplished by rotating S t (and
the head cutter) about the Yb-axis by the tilt angle i (fig. 4(b)}.
The gear settings (fig. 3) are determined with the following parameters:
Em (the machine offset); Ym (the machine root angle); &B (the sliding base);
and _A (the machine center-to-back distance) (refs. 4 and 5).
The head cutter settings (fig. 2) are determined with the following param-
eters: SR (radial setting); q (cradle angle), j (swivel angle); i (tilt angle,
not shown in fig. 2). Parameter q is a variable, since the cradle rotates
during tooth generation but is constant for Formate 1 cut gears.
COLLISION OF HEAD CUTTER WITH STRADDLE GEAR SHAFT BEING DESIGNED
A circle of radius r c on the head cutter (fig. 4) generates a surface in
the coordinate system S while the cradle with the head cutter rotates about
the Zo-axis (fig. 2). _his circle may be represented in coordinate system S t
in parametric form as follows (fig. 4):
r t = rc[COS 0 t sin 0 t 0 1] T (1)
The transposed column matrix represents the homogeneous coordinates of a point
on circle r c.
1Gleason Works trademark.
The generated surface is represented in coordinate system Sp by the
matrix equation
r_t)(Ot,q) = [Mpn][Mno][Moc][Mcb][Mbt]rt(Ot) (2)
Matrix [_nO] describes coordinate transformation from SO to an auxiliary
coordinate system Sn (fig. 3) whose axes are parallel to the axes of SO .
(1)(Ot,q) designates surface number 1 (the head cut-The superscript in rp
ter surface) that is represented in Sp. The surface of the shaft in Sp will
be designated number 2. Here
_'tMbtJ = -si i 0 cos iL: o o
[3)
[Mcb] =
where
-cos(90 o _ _)
cos S
0
0
_ = 21r- j
cos_o°- _ o _1cos(90 ° - S) 0
0 0
cos q
-sin q
[Moc] = o
0
sin q
cos q
0
0
"-sin j
COS j
0
0
0t01
0
-COS j
-sin j
0
0
0
0
1
0
(4)
(5)
Ii 0 0 0 1
1 0 Em
[Mno] = 0 1 - B
o o _
(6)
I -i1cos Ym 0 sin Ym0 1 0[MPn] = -sin Ym 0 cos Ym
0 0 0
(7)
By using equations (1) to (7), the head cutter surface Z1 can be represented
as
3
,q)p -- fl(Ot
y(1) f2 'q)p = (0 t
z(1) f3 'q)p = (0 t
(8)
Now consider a family of coaxial cylinders in coordinate system
radius r i {fig. 5) whose equations are represented by
x (2) - u
p s
y_2) . ri cos Os
Z_ 2) = r i sin 0 s
Sp of
(9)
These coaxial cylinders represent gear shafts of radius r i.
Positive and negative signs on u s correspond to the projection of cur-
rent cylinder point M on the positive or negative axis X The origin of
coordinate system SD coincides with the intersection of t_e shortest distance
with the pinion axis-(for a hypoid gear drive) or with the apex of the pitch
cone (for a spiral bevel gear drive).
The collision of the head cutter surface
means that the surfaces have a point in common.
represented by the vector equation
E1 with the shaft surface E2
Thus, the collision can be
rp(1)(Ot,q) = rp{2)(Os,U s) (_o)
which yields
fl(Ot,q) = u s [
f2(Ot,q) = r i cos Os I (11)f3(Ot,q} = r i sin 0 s
The goal is to determine a subspace in the space of parameters (ri,u s) that is
free from collision of X1 and E2. The determination of such a subspace is
based on the following computational procedure:
Step 1: Parameter q, as mentioned previously is a variable parameter,
and its range of variation must be determined. Considering that qo is the
initial value of q, the limiting values of q, ql and q2, can be determined
from the equations
"N
ql = qo - aq l
Jq2 = qo + _q
4
(12)
where
(13}
_q _ Nll sin yp
yp is the pitch angle of the gear, and N 1 is the number of gear teeth.
Step 2: A certain value for q must be chosen in the range ql £ q _ q2
and then the value of e t varied in the range 0 £ e t £ 2v. The direct compu-
tation allows determination of r i cos e s and r i sin e s from equation (11).
Then r i is obtained by taking into account that
[(r i cos 0s)2 + (r i sin 0s)2] 1/2 = r. (14)I
Equation (10) is used to determine the values of u s that correspond to 0 s.
Step 3: The described computational prqcedure permits a numgrical solu-
tion of a family of closed curves ri[us,q/ij] (fig. 6), where q[i) are the
fixed values of q. A current point N on each curve of the family of curves
in determined with the couple ""._]q(i)'o_J)]' _ where q(i) is a constant and O_j"
)
is a variable. The subspace in the coordinate system (us,r i) that is free
from the family of curves determines the values of r i and u s that can be
chosen by the designer to avoid collision between the shaft and the head cut-
ter. Recall that the origin of Sp coincides with the intersection of the
shortest distance with the pinion axis (hypoid) or with the apex of the pitch
cone (spiral bevel). A computer program for determining the subspace that is
free from collision between the head cutter and the shaft has been developed.
LIMITING LINES OF SPACE rs,U s
The limiting lines of the space rs,U s are shown in figure 7 for a hypoid
pinion. Segments L1 and L2 of the limiting lines in figure 7 represent the
envelopes of the family of closed curves. Segment L3 belongs to the curve
with the initial and final value of q. Segment L4 (fig. 8) represents the
limiting line that corresponds to tangency of the bottom of the head cutter
(fig. 4) with the set of cylinders of radii r i (fig. 5}. Those contact lines
are shown in figure 8.
The limiting lines for the earlier discussion is shown in figure 9. These
lines determine the border of the free space where the extended shaft of the
pinion can be located. Figure 9 indicates that a shaft of 19 mm radius can
have a length of 40 mm.
The derivation of envelopes L 1 and L2 is based on the following ideas:
Consider that the family of planar curves is represented in the shape of param-
eters (u,r) by the equation
R = u(0 t,q)e u + r(0t,q)e r (15)
where q is the parameter of the family. The conditions of envelope existence
are represented by the equation (ref. 2)
Here
al_ f(Ot,q) = 0
Sr 8u ] TN = aet]
(16)
(17)
is the normal to a curve of the family. The tangent T to the curve is rep-
resented by
(18)
T
8_RR [8.._ 8_] (19}8q =
Equations (15) and (16) considered simultaneously represent the envelope of the
family of curves that limits the space (u,r).
The derivation of L 4 is based on the idea that the bottom of the head
cutter (it is a plane) can generate in (u,r) an envelope of a family of planes.
Consider that the bottom of the head cutter is a ring and each circle of the
ring can be represented by its radius r i and a parameter O i. The envelope
generated by the family of rings is represented in (u,r) by the equations
u = u(ri,O i )
r = r(ri'Oi) l (20)r i sin 0 i = SR cos j
The respective values of O i can be obtained by varying r i. As a
reminder, the matrix [MOt], which describes the coordinate transformation from
S t to SO , must be used to derive the functions u(ri,Oi) and r(ri,Oi). Then
the points for the limiting line L 4 may be selected.
CONCLUDING REMARKS
A methodology has been proposed that enables spiral bevel or hypoid gear
teeth with straddle design to be generated without collision between the head
cutter and the shaft. This methodology permits the design of a shaft that
extends toward or beyond the gear apex within a given envelope. Equations are
used to determine the surface generated by the edge of the head cutter. A
zone of safe straddle shaft design can then be determined.
ACKNOWLEDCMENT
This research was partially supported by the Gleason Memorial Fund.
6
APPENDIX - BASIC PINION MACHINE TOOL SETTINGS
The data used in the example of this report for determining the free space
where the gear shaft can be located are given in the following table:
Hypoid gear
Basic tilt angle, deg
Swivel angle, deg
Machine root angle, deg
Cradle angle, deg
Radial setting, ram
Sliding base, mm
Machine center-to-back
distance, mm
Blank offset, mm
Cutting ratio
Cutter point radius, mm
Cutter blade angle, deg
Concave
side
17.260000
-31.266000
357.5160
81.21620
108.9340
19.10001
-2.650000
27.11000
0.2535497
112.7750
14.000000
Convex
side
16.086000
-48.216700
357.5830
72.21670
111.8520
27.64999
3.840000
31.83000
0.2371070
113.4100
31.000000
REFERENCES
1. Drago, R.J.: On the Design and Manufacture of Integral Spiral Bevel Gears
for the CH-47D Helicopter. Presented at the Aerospace Gearing Committee
Meeting, AGMA, Lake Tahoe, NV, Mar. 1979.
2. Litvin, F.L.: Theory of Gearing. NASA RP-1212, 1989.
3. Litvin, F.L.; Zhang, Y.; Lundy, M.; and Heine, C.: Determination of Set-
tings of a Tilted Head Cutter for Generation of Hypoid and Spiral Bevel
Gears. ASME J. Mech. Transmissions Automation Des., vol. 1, Dec. 1988,
pp. 495-500.
4. Litvin, F.L.; and Lee H.-T.: Generation and Tooth Contact Analysis of
Spiral Bevel Gears With Predesigned Parabolic Functions of Transmission
Errors. NASA CR-4259, 1989.
5. Litvin, F.L., et al.: Hypoid Gear Drive With Face-Milled Teeth: Condi-
tions of Pinion Nonundercutting and Fillet Generation. Proceedings of the
1989 Fall Technical Meeting of the American Gear Manufacturers Association,
AGMA, Alexandria, VA, 1989, Paper 7.
"\
\
\
t
i
i
i
"x\ l
"_--Sup_rt for bearing surfaces --3
Figure 1,--Straddle-design spiral bevel gear showing
parameters r and u,
Xb
Xc
Figure 2.--Cutting machine and cradle coordinate systems.
,Y0
o(c)
/
,_Xo
oo /
Em
I
l'IIt"
Zn r_/ . /_
Zp
Figure 3.--Angular velocities of cradle and pinion.
_/--Cutter blade
\ _/_ .n0,.o
YtP' /
Zt
(a) Head cutter surface parameters.
i I-- Original
-' cutter
,,, / orientation
I
co.er',, \ ..!-.,Final
orientation -I _/"', _ X b
Zb Zt
(b) Coordinate systems for head cutter tilt.
Figure 4.--Pinion head cutter surface.
Zp
Figure 5.--Family of coaxial cylinders.
--0-- Initialq rl i_ Initial q
Final q
200 200 /
160 -- 160 I
_ 120_ _ 120_
._ 80 g 80 L
_ 4o .._; 4o- , _,
0 _ us 0
-40 I t 1 _ i _1 1_"'_'_ Ir I -40
-160-120 -80 -40 0 40 80 120 160 200 -160-120 --80 -40 0 40 80 120 160 200
Length, Us, mm Length, Us, rnm
Figure 6.--Family of closed curves. Figure 7._Limiting lines and envelopes.
E
E
Z--
'1:3
80
40m
0 _
-40
--40
m
--o-- Initial q
Final q
80f 2,'3 F',
/ /--L 2
--40
0 40 80 120 160 -120 -80 -40 0 40 80 120
Length, Us, mm Length, Us, mm
Figure 8.--Umiting lines, L4. Figure 9.--Border of free space.
I
160
10
Report Documentation Page
National Aeronautics and
Space Administration
1. Report No. NASA TM-103621
AVSCOM TR 90-C-010
4. Title and Subtitle
7.
9.
112.
2. Government Accession No.
Straddle Design of Spiral Bevel and Hypoid Pinions and Gears
Author(s)
Faydor L. Litvin, Chihping Kuan, Jonathan Kieffer,
Robert Bossier, and Robert F. Handschuh
Performing Organization Name and Address
NASA Lewis Research Center
Cleveland, Ohio 44135-3191
and
Propulsion Directorate
U.S. Army Aviation Systems Command
Cleveland, Ohio 44135-3191
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
3. Recipient's Catalog No.
5. Report Date
November 1990
6. Performing Organization Code
8. Performing Organization Report No.
E-5467
10. Work Unit No.
505-63-51
1L162211A47A
11. Contract or Grant No.
13. Type of Report and Period Covered
Technical Memorandum
14. Sponsoring Agency Code
15. Supplementary Notes
Faydor L. Litvin, Chihping Kuan, and Jonathan Kieffer, University of Illinois at Chicago, Chicago, Illinois 60680.
Robert Bossier, Lucas Western Inc., City of Industry, California; and Robert F. Handschuh, Propulsion
Directorate, U.S. Army Aviation Systems Command.
16. Abstract
The design of spiral bevel and hypoid gears that have a shaft extended from both sides of the cone apex (straddle
design) is considered. A main difficulty of such a design is determining the length and diameter of the shaft that
might be undercut by the head cutter during gear tooth generation. A method that determines the free space
available for the gear shaft is proposed. The approach avoids collision between the shaft being designed and the
head cutter during tooth generation. The approach is illustrated with a numerical example.
17. Key Words (Suggested by Author(s))
Gears; Spiral bevel gears; Gear geometry; Gear
manufacture; Straddle gear design
18. Distribution Statement
Unclassified - Unlimited
Subject Category 37
19. Security Classif. (of this report) 20. Security Classif. (of this page) 21. No. of pages
Unclassified Unclassified 12
NASA FORM 1626 OCT 86
*For sale by the National Technical InformationService, Springfield, Virginia 22161
22. Price*
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Straddle design of spiral bevel and hypoid pinions and gears

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