ISO/TR 22849:2011
(Main)Design recommendations for bevel gears
Design recommendations for bevel gears
ISO/TR 22849:2011 provides information for the application of bevel and hypoid gears using the geometry in ISO 23509, the capacity as determined by ISO 10300 (all parts) and the tolerances in ISO 17485. ISO/TR 22849:2011 provides additional information on the application, manufacturing, strength and efficiency of bevel gears for consideration in the design stage of a new bevel gear set.
Recommandations pour le dimensionnement d'engrenages coniques
General Information
Standards Content (Sample)
TECHNICAL ISO/TR
REPORT 22849
First edition
2011-04-15
Design recommendations for bevel gears
Recommandations pour le dimensionnement d'engrenages coniques
Reference number
ISO/TR 22849:2011(E)
©
ISO 2011
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ISO/TR 22849:2011(E)
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ISO/TR 22849:2011(E)
Contents Page
Foreword .iv
1 Scope.1
2 Symbols, descriptions and units.1
3 Application .3
3.1 Geometry.3
3.2 Rating.3
3.3 Materials .4
3.4 Gear tolerances .4
3.5 Gear noise .4
4 Manufacturing consideration .8
4.1 Outline of production methods and their features — Face milling and face hobbing.8
4.2 Blank design and tolerances.9
4.3 Assembly.18
4.4 Tooth contact pattern.23
5 Strength considerations .25
5.1 Effect of hypoid offset.25
5.2 Effect of cutter radius .25
5.3 Bevel gear mountings.27
5.4 Direction of forces.29
6 Efficiency considerations.29
6.1 Hypoid and bevel gear mesh efficiency.29
6.2 Lubrication .33
Bibliography.37
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ISO/TR 22849:2011(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through
ISO technical committees. Each member body interested in a subject for which a technical committee has
been established has the right to be represented on that committee. International organizations, governmental
and non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
In exceptional circumstances, when a technical committee has collected data of a different kind from that
which is normally published as an International Standard (“state of the art”, for example), it may decide by a
simple majority vote of its participating members to publish a Technical Report. A Technical Report is entirely
informative in nature and does not have to be reviewed until the data it provides are considered to be no
longer valid or useful.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO/TR 22849 was prepared by Technical Committee ISO/TC 60, Gears, Subcommittee SC 2, Gear capacity
calculation.
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TECHNICAL REPORT ISO/TR 22849:2011(E)
Design recommendations for bevel gears
1 Scope
This Technical Report provides information for the application of bevel and hypoid gears using the geometry in
ISO 23509, the capacity as determined by ISO 10300 (all parts) and the tolerances in ISO 17485.
This Technical Report provides additional information on the application, manufacturing, strength and
efficiency of bevel gears for consideration in the design stage of a new bevel gear set.
The term “bevel gear” is used to mean straight, spiral, zerol bevel and hypoid gear designs. Where this
Technical Report pertains to one or more, but not all, the specific forms are identified.
The manufacturing process of forming the desired tooth form is not intended to imply any specific process, but
rather to be general in nature and applicable to all methods of manufacture.
This Technical Report is intended for use by an experienced gear designer capable of selecting reasonable
values for the required data based on his/her knowledge and background. It is not intended for use by the
engineering public at large.
2 Symbols, descriptions and units
The symbols and descriptions used in this Technical Report are, wherever possible, consistent with other
International Standards on bevel gears. As a result of certain limitations, some symbols and descriptions are
different than in similar literature pertaining to spur and helical gearing.
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ISO/TR 22849:2011(E)
Symbol Description Unit
A Arrangement constant —
g
a Centre distance of virtual cylindrical gears mm
v
α Generated pressure angle according to ISO 23509 °
n
α , α Pressure angle at tip of virtual cylindrical gear °
vat1 vat2
α Pressure angle in transverse plane of virtual cylindrical gear °
vt
b Face width in contact with mating element mm
eff
β Outer spiral angle according to ISO 23509 —
e
β , β Mean spiral angle °
m1 m2
β Spiral angle of virtual cylindrical gear °
v
C A constant —
1
D Outside diameter of the considered rotating element mm
d , d Outside diameter mm
ae1 ae2
d , d Reference diameter of virtual cylindrical gear mm
v1 v2
d , d Tip diameter of virtual cylindrical gear mm
va1 va2
Δα , Δα Change in pressure angle from pitch point to outside °
t1 t2
δ , δ Face angle °
a1 a2
δ , δ Pitch angle °
1 2
ΣP Sum of power losses regarding churning kW
GW
f Gear dip factor —
g
ϕ Friction angle °
h , h Mean addendum mm
am1 am2
η Sum of element churning efficiency (see 6.1.6) —
ffc
η Lengthwise sliding efficiency (see 6.1.5) —
ffl
η Profile sliding efficiency (see 6.1.4) —
ffp
j Outer transverse backlash mm
et
j Outer normal backlash mm
en
2
K Load intensity for calculating the coefficient of friction N/mm
L Length of the element of the considered rotating element mm
m Transverse tooth module of the gear considered mm
t
μ Coefficient of friction (see 6.1.6) —
m
n Rotational shaft speed r/min
2
υ, υ Kinematic oil viscosity at operating temperature, kinematic viscosity at 40 °C mm /s (cSt)
κ
v Pitch line velocity at outside diameter m/s
et
P Design power kW
P Power loss for each individual element kW
GWi
R , R Mean cone distance mm
m1 m2
R Roughness factor —
f
T Output torque, wheel, per unit force mm
o2
T , T Input torque per unit force, pinion and wheel mm
i1 i2
t , t Pinion back angle distance mm
B1 B2
t , t Pinion crown to back mm
E1 E2
t , t Pinion face angle distance mm
F1 F2
T Pinion torque Nm
1
z , z Number of pinion, wheel teeth —
1 2
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ISO/TR 22849:2011(E)
3 Application
3.1 Geometry
For the purposes of this Technical Report, the geometry of bevel and hypoid gear pairs is assumed to be
calculated according to ISO 23509. These calculations need at least a set of initial data. If these data are not
completely given or known from similar applications, a rough estimate of the gear dimensions can be
determined by means of the power to be transmitted (see Annex B of ISO 23509:2006).
In any case, a complete geometry calculation has to be successfully executed before any other of the
following considerations makes sense.
3.2 Rating
3.2.1 General
To make a rating of a pair of bevel gears one should have a mathematically correct set of geometry (see 3.1).
This enables the designer to proceed to more detailed calculations which complete the design insofar as the
transmitted torque is concerned. Additional rating criteria for bending strength and pitting resistance should
also be considered. The method for calculating the bending strength and pitting resistance of bevel gears
except hypoid gears is stated in ISO 10300 (all parts).
3.2.2 Bending strength
Bending strength as a criterion of bevel and hypoid gear capacity can be defined as the ability of the gear set
to withstand repeated or continued operation under nominal load without fracture of the teeth in their roots by
fatigue in bending. It is a function of the bending (tensile) stresses in a cantilever beam and is proportional to
the applied load. It also involves the fatigue strength of the gear materials and the shape of the teeth.
Therefore, either the pinion or the wheel can be the limiting member of the pair.
3.2.3 Pitting resistance
Pitting resistance as a criterion of bevel and hypoid gear capacity can be defined as the ability of the gear set
to withstand repeated or continued operation under nominal load without suffering destructive pitting of the
tooth surfaces. The experienced gear designer recognizes that moderate, non-destructive pitting of the tooth
surfaces can occur during the early stages of operation, especially on non-hardened or through-hardened
gears. In these cases, the pitting ceases to progress after the asperities have been removed by the initial
operation. This process, called initial pitting, should not affect the gear life.
Destructive pitting, although attributable in principle to the same phenomena, progresses widely enough to
destroy the geometry of the flank surfaces and ultimately leads to failure. The distinction between initial and
destructive pitting is defined more thoroughly in ISO 10825.
Pitting is a function of several factors; the most significant is Hertzian contact (compressive) stresses between
the two mating tooth surfaces and is proportional to the square root of the applied tooth load. The ability of
bevel and hypoid gear teeth to withstand repeated surface contact under load without destructive pitting
involves the resistance of the gear material to fatigue under contact stresses. The smaller gear is usually the
limiting member of the pair because the teeth receive more stress cycles per unit time. In some cases, the
smaller gear is made harder than its mate, to increase its surface durability so that the limiting capacity can
exist in either member.
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ISO/TR 22849:2011(E)
3.2.4 Other forms of bevel gear tooth deterioration
The rating standards are not applicable to other types of gear tooth deterioration such as micropitting, case
crushing, wear, plastic yielding and welding.
Information on scuffing can be found in ISO/TR 13989-1.
3.3 Materials
The quality of materials and methods of heat treatment required are governed by the application. Care should
be taken to choose the proper material for each application to transmit the load and obtain the life desired.
Heat treatment is usually needed to develop the necessary hardness, strength and wear resistance.
For information about materials and heat treatment, see ISO 6336-5.
3.4 Gear tolerances
Bevel gears are manufactured to suit many engineering applications. In order to satisfy these needs properly,
it is necessary to analyse the conditions under which these gears should operate. Reasonable tolerances
should then be established to ensure that the gears perform satisfactorily in the application.
Tolerance values for unassembled bevel gears, hypoid gears and gear pairs are provided in ISO 17485.
Additionally, information about bevel gear measurement methods is given in ISO/TR 10064-6.
3.5 Gear noise
3.5.1 General
The gear noise can be produced by the vibration of the gear unit caused by the transmission error of the gear
pair. The flank form deviations of the teeth, a misalignment between the gears, and the elastic deformation of
the teeth under load affect the transmission error. Table 1 shows typical values of transmission errors for
different gear applications.
Table 1 — Typical values of transmission error
Recommendation value
Application
μrad
Passenger car <30
Truck 20 to 50
Industrial 40 to 100
Aircraft 40 to 200 (80 average)
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ISO/TR 22849:2011(E)
3.5.2 Tooth flank form corrections
The tooth flank form of bevel gears is corrected in order to prevent edge contact of tooth bearing during
operation. Figure 1 a) shows the tooth flank form deviation of a spiral bevel gear after lapping. The amount of
deviation between adjoining contour lines is 2 μm. It turns out that a crowning of remarkable size occurs in
face width direction. On the other hand, the amount of deviation in the profile direction is small. Figure 1 b)
shows the pertaining tooth bearing and Figure 1 c) the waveform of the transmission error. The peak-to-peak
value of the transmission error is 24 μrad.
a) Tooth flank deviation b) Tooth bearing
c) Transmission error
Figure 1 — Example of a gear pair finished by lapping process
Figure 2 shows the effect of profile crowning and flank twist where the amount of lengthwise crowning is fixed
at 20 μm. In the case of 5 μm profile crowning in Figure 2 a), the width of tooth bearing is wide, and the
transmission error is 27 μm. On the other hand, in the case of 20 μm profile crowning in Figure 2 b), the width
of tooth bearing is narrow, and the transmission error increases to 43 μm. This means that excessive profile
crowning should be avoided. However, in the case of Figure 2 c) with a flank twist correction of 80 μm, the
transmission error decreases to 24 μrad. This shows the effectiveness of flank twist modifications if the profile
crowning is enlarged.
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ISO/TR 22849:2011(E)
NOTE The transmission error is 27 μrad. NOTE The transmission error is 43 μrad.
a) Profile crowning of 5 μm b) Profile crowning of 20 μm
NOTE The transmission error is 24 μrad.
c) Profile crowning at 20 μm and flank twist of 80 μm
Figure 2 — Effect of profile crowning and flank twist on transmission error —
Lengthwise crowning of 20 µm
Since tooth flanks are subject to elastic deformations under load, this needs to be considered for flank form
corrections. However, as the noise of a gear set in many cases becomes a problem under light load, the
measure indicated above is rather effective.
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ISO/TR 22849:2011(E)
3.5.3 Design contact ratio
NOTE The transmission error is 24 μrad. NOTE The transmission error is 52 μrad.
a) Number of teeth of 10/43 and b) Number of teeth of 7/30 and
contact ratio of 3:02 contact ratio of 2:23
Figure 3 — The effect of design contact ratio on transmission error
The design contact ratio is the angle of transmission of one pair of teeth divided by the angular pitch. It is
favourable, therefore, to enlarge the design contact ratio of a gear set in order to reduce gear noise. To get a
higher design contact ratio, it is effective to increase the number of teeth, to enlarge the working tooth depth
and to increase the spiral angle. However, if the number of teeth is increased, the mean normal module
becomes smaller and reduces the load carrying capacity. Moreover, there is a risk that undercut can occur in
the pinion root or the topland can become too small if the tooth depth is enlarged too much. Caution is
required in those points.
Figure 3 shows the effect of design contact ratio on the transmission error. In the case of smaller tooth
numbers of 7/30, the contact ratio which is less than that of 10/43, the size of the tooth bearing increases, but
the transmission error also increases from 24 µrad to 52 µrad, although the tooth flank deviations are the
same.
The actual contact ratio can change under load by deformation of the flanks and deflections of teeth and
shafts.
3.5.4 Other noise consideration
Where a large misalignment is in the mountings of a gear pair or the misalignment produced by deflection
under load is considerable, the tooth flanks can have edge contact and the transmission error can increase.
Therefore, caution is advised to make the mountings of the gears accurate and the rigidity of the gearbox
high.
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ISO/TR 22849:2011(E)
4 Manufacturing consideration
4.1 Outline of production methods and their features — Face milling and face hobbing
In principle, there are two different methods used for manufacturing spiral bevel gears and hypoid gears:
single indexing, which is also called face milling (FM) and continuous indexing, which is also called face
hobbing (FH).
For the FH method, the rotation of the tool and of the workpiece are coupled in a fixed ratio so that one blade
group of the cutter head enters one tooth gap and the next blade group enters the next gap, etc. This method
is called continuous indexing, where all gaps are cut at the same time and which produces an epicycloid in the
lengthwise direction. Generally, the tooth depth is constant along the face width so that root angle and face
angle are equal. The tooth geometry results in a tapered topland and a tapered slot width. With a reasonably
sized cutter radius, the tooth gap at the toe is slightly smaller than at the heel. If the cutter radius is too small,
the inner tooth end becomes thicker than the outer. Therefore, too small cutter radii should be avoided.
In the FM method, the cutter blades are set in a circle on the cutter head. This method is used for cutting and
for grinding. The tooth gaps are manufactured by single indexing which means that one gap is finish cut (or
ground), then the gear blank is rotated by one pitch and the next gap is cut. Consequently, this method
produces a circular arc in tooth lengthwise direction and in its standard geometry the tooth depth is tapered as
well as topland and slot width. However, if root angle and face angle are specifically changed by a tilted root
line depending on the cutter radius, the tapered tooth gets constant slot width and nearly constant topland,
while maintaining proper space width taper (see 5.3.2.1 of ISO 23509:2006). The advantage of this measure
is that FM bevel gears can also be completely cut in one single clamping.
Although there is an obvious difference between face hobbed gears and face milled gears, it does not lead to
a general rule that one or the other method gives better results. The only fact is that for hardened spiral bevel
gears no grinding process exists with continuous indexing, but instead precision hard cutting. Moreover, for
bevel gears with diameters of more than 1 000 mm, there is no other way than continuous hard cutting
because such big grinding machines are not available.
Regarding operating properties and load carrying capacity, no difference between FH and FM bevel gears can
be found, if all crucial parameters are kept the same. These findings are also promoted by modern tooth
contact analyses and FM calculation programs by which bevel gear designs can be checked and optimized.
These programs also allow the study of detailed tooth flank modifications prior to manufacturing.
Historically, the choice of the manufacturing method was determined by the cutting machine available from a
particular distributor. Nearly all new machines are 6-axes CNC machines, which can realize both face milling
as well as face hobbing, and most of the current submethods.
Any heat treatment distortions are independent of the FH or FM method. With small distortions, lapping is the
usually applied finishing process which also works equally with FH and FM bevel gears. However, in the case
of larger distortions, a grinding or cutting process is required. Then, it is obvious to use grinding for FM gears
and hard cutting for FH gears as their respective geometries are identical. Face hobbed gears can also be
ground, however by single indexing, and both flanks should be ground separately for a correct engagement.
Unfortunately, hardening distortions hinder a geometrically stable lapping process. If these distortions are
known exactly in advance, the flank form can be modified in the cutting process to compensate for the
distortion so that the lapping process can be used more effectively. Contrary to grinding and hard cutting,
lapping is a process without high geometric consistency but with the advantage of less noise emission by
reducing relative tooth profile error between pinion and wheel, if heat treatment distortion is limited. Lapping
does not eliminate any deviation in pitch and runout. Generally, the lapping process cannot be used to apply
any designed flank or profile modification. This is possible with grinding and precision hard cutting only.
The cutting method and finishing method that should be used depends on the intended use of the respective
gears, the equipment available and a lot of other aspects.
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ISO/TR 22849:2011(E)
4.2 Blank design and tolerances
4.2.1 General aspects
The quality of any finished gear is dependent on the design and accuracy of the gear blank. A number of
important factors which affect cost, as well as performance, should be considered.
Bores, hubs, and other locating surfaces should be in proper proportion to the gear diameter and module.
Small bores, thin webs, and any condition that results in excessive overhang and deflection, should be
avoided.
4.2.2 Clamping surface
Nearly all bore-type bevel gears are held by means of a clamp plate at the front face of the hub when the teeth
are being cut; therefore, the blank should incorporate a suitable surface for this purpose, as shown in
Figure 4.
Key
1 no surface provided for clamping, not recommended
2 clamping surface, as recommended
Figure 4 — Recommended clamping surface of the blank
4.2.3 Tooth backing
Sufficient thickness of metal should be provided under the roots of gear teeth to give proper support for the
teeth. It is suggested that the minimum amount of material under the teeth not be less than the whole depth of
the tooth. Highly stressed gears can require additional backing. This material depth should be maintained
under the small ends of the teeth as well as under the middle (see Figure 5). In addition, on webless-type
wheels the minimum stock between the bottom of the tap drill hole and the gear root line should be one third
the tooth depth.
Key
1 tooth backing, as recommended
Figure 5 — Tooth backing
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ISO/TR 22849:2011(E)
4.2.4 Load direction
A gear blank should be designed to avoid excessive localized stresses and serious deflections within itself.
For heavily stressed gears, a preliminary analysis of the direction and magnitude of the forces is helpful in the
design of both the gear and the mounting. Where possible, the direction of the web should coincide with the
direction of the resultant tooth load in an axial section. Gear sections should be designed in such a way that a
component of the tooth load is directed through the section as shown in Figure 6. See ISO 23509:2006,
Annex D, for detailed discussions of tooth loads.
Key
1 tooth load component
Figure 6 — Webless mitre gear — Counterbored type
4.2.5 Locating surface
The back of the gears should be designed with a locating surface of generous size. This surface should be
machined or ground square with the bore and is used both for locating the gear axially in assembly and for
holding it when the teeth are cut. The front clamping surface should, of course, be flat and parallel to the back
surface. A flat and parallel surface also provides a convenient inspection surface after installation.
Gears with a comparatively large ratio of pitch diameter to hub diameter, greater than 2,5:1, should have an
auxiliary locating surface behind the teeth as shown in Figure 7. A similar surface should also be used for thin-
webbed gears where there is danger of blank distortion or vibration from cutting forces.
Key
1 suggested locating surfaces
Figure 7 — Suggested locating surfaces
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ISO/TR 22849:2011(E)
4.2.6 Solid shanks
Where gears with solid shanks are made in large quantities, a collet chuck is usually used. For small
quantities, the gears should be provided with a tapped hole or external threads at the end of the shank to hold
the gear securely in the chuck while cutting the teeth (see Figures 8 and 9).
Figure 8 — Shank-type pinion with tapped hole
Key
1 centres (as large as possible and relieved as shown)
Figure 9 — Shank-type pinion with external threads
4.2.7 Flanged hub
Whether the gear is mounted on a flanged hub or is made integral with the hub, the supporting flange should
be of sufficient section size to prevent deflections in the direction of the gear axis at the mesh point.
The web preferably should be made conical without ribbing to permit rough machining of the blanks for
obtaining better balance, to eliminate oil churning when dip lubrication is used, and lessen the danger of
stress concentration being set up within the castings.
4.2.8 Splined bores
In mounting gears with splined bores, a piloting diameter is suggested to reduce eccentricity. Hardened gears
with straight-sided splines in the bore should be piloted in assembly by the bore or minor diameter of the
splines, which should be ground concentric with the teeth after hardening. Unhardened gears with straight-
sided splines should be piloted in assembly by the major diameter of the splines. In either case, the finish
machining of the blank, cutting of teeth and the soft testing should be performed with the gear centred on the
arbor by the bore, which has been machined true with the splines.
Figure 10 shows a gear with a cylindrical fit at each end of the bore, the splines being used for driving only.
This type of fit is particularly applicable to aircraft gears, which often use involute splines with a full fillet radius
on the major diameter. This design is an excellent solution, particularly when the splines have to be hardened,
because fitting on the sides of the splines is extremely difficult when size changes and distortion takes place
during heat treatment.
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