Chapter 9 Spur Gear Design

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Chapter 9 Spur Gear Design
Chapter 9 Spur Gear Design
The big picture: What to learn in chapter 9?
• A spur gear has involute teeth that are straight and
parallel to the axis of the shaft that carries the gear.
1. Describe the action of the teeth of the driving gear on those
of the driven gear.
2. what kind of stresses are produced?
3. How do the geometry of the gear teeth,the materials from
which they are made , and the operating conditions affect
the stresses and the life of the drive system?
About a spur gear
• A spur gear is one of the most fundamental types
of gears.
• Its teeth are straight and parallel to the axis of the
shaft that carries the gear. The teeth have the
involute( 渐开线) form.
• The action of one tooth on a mating tooth is like
that of two convex(凸轮廓),curved members in
contact: as the driving gear rotates,its teeth exert a
force on the mating gear that is tangential to the
pitch circles of the two gears.
Consider the action described in the preceding paragraph:
• How does that action related to the design of the
gear teeth? Figure 8-1
• As the force is exerted by the driving tooth on the
driven tooth, what kinds of stresses are produced
in the teeth? Consider both the point of contact of
one tooth on the other and the whole tooth. Where
are stresses a maximum?
• How could the teeth fail under the influence of
these stresses?
• What material properties are critical to allow the gears to
carry such loads safely and with a reasonable life span?
• What are the important geometric features that affect the
level of stress produced in the teeth?
• How does the precision of the tooth geometry affect its
• How does the nature of the application affect the gears?
For example,what if the machine that the gears drive is a
rock crusher (碎石机)that takes large boulders(大石头)
and reduces them to gravel(砂砾) made up of small stones?
How would that loading compare with that of a gear
system that drives a fan providing ventilation air(流通空气)
to a building?
• What is the influence of the driving machine?
Would the design be different if an electric motor
were the driver or if a gasoline engine were used?
• The gears are typically mounted on shafts that
deliver power from the driver to the input gear of a
gear train and that take power from the output gear
and transmit it to the driven machine.Describe
various ways that the gears can be attached to the
shafts and located with respect to each other. How
can the shafts be supported?
You are the designer
• The teeth must not break;
• They must have a sufficiently long life to
meet the needs of the customer who uses
the reducer.
We need more data
• How much power is to be transmitted?
• To what kind of machine is the power from the
output of the reducer being delivered?
• How does that affect the design of the gears?
• What is the anticipated duty cycle for the reducer
in terms of the number of hours per day,days per
week,materials that are suitable for gears?
• Which material will you specify , and what will
be its heat treatment?
9-1 objectives of this chapter
After completing this chapter,you can do demonstration of
competencies as following:
1. Compute the forces exerted on gear teeth as they rotate and
transmit power ;
2. Describe various methods for manufacturing gears and the levels
of precision and quality to which they can be produced;
3. Specify a suitable level of quality for gears according to the use
to which they are to be put;
4. Describe suitable materials from which to make the
gears, in order to provide adequate performance for both
strength and pitting resistance;
5. Use the standards of the American Gear
Manufacturers Association (AGMA) as the basis for
completing the design of the gears;
6. Use appropriate stress analyses to determine the
relationships among the applied forces, the geometry of
the gear teeth,the precision of the gear teeth, and other
factors specific to a given application, in order to make
final decisions about those variables.
9-1 Objectives of this chapter
7. Perform the analysis of the tendency for the contact
stresses exerted on the surfaces of the teeth to cause pitting
of the teeth,in order to determine an adequate hardness of
the gear material that will provide an acceptable level of
pitting resistance for the reducer ;
8. Complete the design of the gears,taking into consideration
both the stress analysis and the analysis of pitting
resistance. The result will be a complete specification of
the gear geometry,the material for the gear, and the heat
treatment of the material.
9-2 concepts from previous chapters
• As learned in chapter 8, key relationships that you
should be able to use include the following:
pitch line speed   t  R  ( D / 2)
where R  radius of the pitch circle
D  pitch diameter
  angular ve locity of the gear
Because the pitch line speed is the same for both the pinion
and the gear, value for R, D, and  can be for either. In the
computation of stresses in gear teeth, it is usual to express the
pitch line speed in the units of ft/min,while the size of the
gear is given as its pitch diameter expressed in inches. Speed
of rotation is typically given as n rpm,that is , n rev/min.
Let’s compute the unit-specific equation that gives pitch
line speed in ft/min:
D in n rev 2 rad 1 ft
vt  ( D / 2) 
rev 12 in
 (Dn / 12) ft / min
(9 - 1)
Velocity ratio
The velocity ratio can be expressed in many ways.
For the particular case of a pinion (小齿轮)
driving a larger gear.
 p n P RG DG N G
velocity ratio  VR 
 G nG R P DP N P
(9 - 2)
Gear ratio
A related ratio,mG,called the gear ratio, is often used in
analysis of the performance of gears. It is defined as the ratio
of the number of teeth in the larger gear to the number of the
teeth in the pinion,regardless of which is the driver. Thus, mG
is always greater than or equal to 1.0. When the pinion is the
driver, as it is for a speed reducer, mG is equal to VR. That is,
Gear ratio  m G  N G / N P  1.0
(9 - 3)
The pressure angle,, is an important feature that characterize the
form of the involute curve that makes up the active face of the
teeth of standard gears. See Fig.8-13,Fig.8-12.
That angle between a normal to the involute curve and the
tangent to the pitch circle of a gear is equal to the pressure angle.
9-3 Forces on gear teeth
• To understand the method of computing stresses in gear
teeth, consider the way power is transmitted by a gear
Torque  power/rota tional speed  P
(9 - 5)
The input shaft transmits the power from the coupling to
the point where the pinion is mounted. The power is
transmitted from the shaft to the pinion through the key.
The teeth of the pinion drive the teeth of the gear and thus
transmit the power to the gear. But again, power
transmission actually involves the application of a torque
during rotation at a given speed.
The torque is the product of the force acting tangent to the
pitch circle of the pinion times the pitch radius of the
we use the symbol Wt to indicate the tangential force.
As described,Wt is the force exerted by the pinion teeth
on the gear teeth.
But if the gears are rotating at a constant speed and are
transmitting a uniform level of power, the system is in
equilibrium. Therefore, there must be an equal and
opposite tangential force exerted by the gear teeth back
on the pinion teeth. This is an application of the
principle of action and reaction.
• To complete the description of the power flow, the
tangential force on the gear teeth produces a
torque on the gear equal to the product of Wt
times the pitch radius of the gear. Because Wt is
the same on the pinion and the gear, but the pitch
radius of the gear is larger than that of the
pinion,the torque on the gear (the output torque) is
greatest than the input torque. However, note that
the power transmitted is the same or slightly less
because of mechanical inefficiencies. The power
then flows from the gear through the key to the
output shaft and finally to the driven machine.
• Gears transmit power by exerting a force by the driving
teeth on the driven teeth while the reaction force acts back
on the teeth of the driving gear.
• Fig.9-2 shows that, a single gear tooth with the tangential
force Wt acting on it. But this is not the total force on the
tooth. Because of the involute form of the tooth, the total
force transferred from one tooth to the mating tooth acts
normal to the involute profile. This action is shown as Wn.
The tangential force Wt is actually the horizontal
component of the total force. To complete the picture, note
that there is a vertical component of the total force acting
radially on the gear tooth, indicated by Wr.
Diametral pitch
• The diametral pitch,Pd,characterizes the physical
size of the teeth of a gear. It is related to the pitch
diameter and the number of teeth as follows:
Pd  N G / DG  N P / DP
(9 - 4)
• Transmitted force,Wt, is based on the given data for
power and speed. It is convenient to develop unitspecific equations for Wt because standard practice
typically calls for the following units for key
quantities pertinent to the analysis of gear sets:
• Forces in pounds (lb);
• Power in horsepower (hp) (Note that 1.0hp =
• Rotational speed in rpm, that is , rev/min
• Pitch line speed in ft/min;
• Torque in lbin
• The torque exerted on a gear is the product of the
transmitted load, Wt, and the pitch radius of the gear. The
torque is also equal to the power transmitted divided by the
rotational speed. Then
T  Wt ( R)  Wt ( D / 2)  P / n
2 P(hp)
550lb  ft / s 1.0 rev 60 s/min 12 in
Wt 
Dn D(in )  n(rev / min)
2 rad
Wt  (126000)( P) /( nD)lb
(9 - 6)
Power is also the product of the transmitted force, Wt, and the
pitch line velocity:
P  Wt  vt
Then, solving for the force and adjusting units,
550 lb / s 60 s/min 12 in
Wt  
vt Vt ( ft / min) 1.0 hp
also compute torque in lb  in :
550 lb  ft/s 1.0 rev 60 s/min 12 in
T 
 n(rev / min)
1.0 hp
2 rad
T  63000( P) / n lb  in
(9 - 7)
(9 - 8)
• These values can be computed for either the pinion or the
gear by appropriate substitutions.
• The pitch line speed is the same for the pinion and the gear;
• The transmitted loads on the pinion and the gear are the
same, except that they act in opposite directions.
The normal force,Wn, and the radial force,Wr, can be
computed from the known Wt by using the right triangle
relations evident in Fig.9-2.
Wr  Wt tan 
(9 - 9)
Wn  Wr / cos 
(9 - 10)
where   pressure angle of the tooth form
analysis of Forces on spur gear teeth
• These forces cause the stresses in the gear teeth;
• These forces act on the shaft; in order to maintain
equilibrium, the bearings that support the shaft must
provide the reactions. Fig.9-1 shows the free-body diagram
of the output shaft of the reducer.
9-4 Gear Manufacture
• Small gears are frequently made from wrought plate or
bar,with the hub,web,spokes,and rim machined to final or
near-final dimensions before the gear teeth are produced.
• The face width and the outside diameter of the gear teeth
are also produced at this stage.
• Other gear blanks may be forged,sand cast to achieve the
basic form prior to machining.
• A few gears in which only moderate precision is required
may be die cast with the teeth in virtually final form.
large gears are frequently fabricated from components.
• The rim and the portion into which the teeth are machined
may be rolled into a ring shape from a flat bar and then
• The web or spokes and the hub are then welded inside the
• Very large gears may be made in segments with the final
assembly of the segments by welding or by mechanical
The popular methods of machining the gear teeth are
form milling(磨削),shaping(修整),and hobbing(齿轮滚铣).
• In form milling(Fig.9-3), a milling cutter(铣刀) that has the
shape of the tooth space is used,and each space is cut
completely before the gear blank(齿轮坯料) is turned to the
position of the next adjacent space. This method is used
mostly for larger gears, and great care is required to achieve
accurate results.
•Shaping (Fig.9-4),is a process in which the cutter
reciprocates,usually on a vertical spindle. The shaping
cutter rotates as it reciprocates and it fed into the gear blank.
Thus, the involute-tooth form is generated gradually. This
process is frequently used for internal gears.
•Hobbing (Fig.9-5,9-6),is a process similar to milling
except that the work-piece (the gear blank,齿轮坯) and
the cutter (the hob) rotate in a coordinated fashion. Here
also,the tooth form is generated gradually as the hob(滚刀)
is fed into the blank.
• The gear teeth are finished to greater precision
after form milling,shaping,or hobbing by the
processes of grinding(磨齿),shaving(剃齿),and
honing(珩磨). Being products of secondary
processes,they are expensive and should be used
only where the operation requires high accuracy in
the tooth form and spacing.
9-5 Gear quality
• Quality in gearing is the precision of the individual
gear teeth and the precision with which two gears
rotate in relation to one another. The factors normally
measured to determine quality are the following:
• Runout(偏转): A measure of eccentricity and out-ofroundness
• Tooth-to-tooth spacing: the difference in spacing between
corresponding points on adjacent teeth;
• Profile: the variation of the actual tooth profile from the
theoretically precise profile.
• Composite error is measured on a special device,shown in
Fig.9-7(a), that places the test gear in tight mesh with a
master gear of known precision. The two gears are rotated
while in tight mesh, and the center of one gear is free to
• The variation in center distance for a complete revolution
of the test gear is recorded.
• Fig.9-7(b) shows a typical recording with both tooth-totooth composite error and total composite error indicated.
Thus, composite error is a measure of the combined effects
of several types of gear-tooth errors.
• The allowable amount of variation of the actual tooth form
from the theoretical form, or the composite error, is
specified by defining an AGMA quality number. Details
charts giving the tolerances for many features are included
in AGMA Standard 2000-A88. The quality numbers range
from 5 to 15 with increasing precision. The actual
tolerances are a function of the diametral pitch of the gear
teeth and of the pitch diameter of the gear.
• Table 9-1 shows representative data for the total composite
tolerance for several quality numbers.
Recommended quality numbers
• The design of the entire gear system,including the shafts,
bearings, and housing, must be consistent with this precision.
• The system should not be made more precise than necessary
because of cost. For this reason, manufacturers have
recommended quality numbers that will give satisfactory
performance at a reasonable cost for a variety of applications.
As shown in Table 9-2. Also shown in the same table, quality
numbers are related to the pitch line speed,defined as the linear
velocity of a point on the pitch circle of the gear.Fig.8-15.
Pitch line speed
Convenient relations for pitch line speed
are adapted from equation(9 - 1) and given here :
v t  DG nG / 12  D p n p / 12
where v t  pitch line speed in feet per minute
n G  rotational speed of the gear (rpm)
n P  rotational speed of the pinion (rpm)
D G  pitch diameter of the gear (in)
D P  pitch diameter of the pinion (in)
(9 - 11)
9-6 Allowable stress numbers
• A gear tooth acts like a cantilever beam in resisting the
force exerted on it by the matting tooth. The point of
highest tensile bending stress is at the root of the tooth
where the involute curve blends with the fillet. The AGMA
has developed a set of allowable bending stress numbers,
called sat,which are compared to computed bending stress
levels in the tooth to rate the acceptability of a design.
A second,independent form of failure is the pitting of the
surface of the teeth, usually near the pitch line,where very
high contact stresses occur. The transfer of force from the
driving to the driven tooth theoretically occurs across a line
contact because of the action of the two convex curves on
each other.
• Repeated application of these high contact stresses can
cause a type of fatigue of the surface,resulting in local
fractures and an actual loss of material. This is called
pitting. The AGMA has developed a set of allowable
contact stress numbers, called sac,which are compared to
computed contact stress levels in the tooth to rate the
acceptability of a design.
9-7 Gear Materials
• Gears can be made from a wide variety of materials to
achieve properties appropriate to the application. From a
mechanical design standpoint,strength and pitting
resistance are the most important properties. But, in
general, the designer should consider the producibility of
the gear,taking into account all of the manufacturing
processes involved, from the preparation of the gear
blank,through the forming of the gear teeth, to the final
assembly of the gear into a machine. Other considerations
are weight,appearance,corrosion resistance,noise, and cost.
Steel gear materials
• Through-hardened steels. Gears for machine tool
drives and many kinds of medium-to heavy-duty
speed reducers and transmissions are typically
made from medium-carbon steels. Among a wide
range of carbon and alloy steels used are
• AISI 1020 AISI 1040 AISI 1050
• AISI 3140 AISI 4140 AISI 4150
• AISI 4340 AISI 6150 AISI 8650
• AGMAStandard 2001-C95 gives data for the allowable
bending stress number,Sat, and the allowable contact stress
number,Sac, for steels in the through-hardened condition.
Fig.9-8 and Fig.9-9 are graphs relating the stress numbers
to the Brinell hardness number for the teeth. Notice that
only knowledge of the hardness is required because of the
direct relationship between hardness and the tensile
strength of steels. See appendix 19 for data that correlate
the brinell hardness number,HB,with the tensile strength of
steel in Ksi. The range of hardness covered by the AGMA
data is from 180 to 400 HB,corresponding to a tensile
strength of approximately 87 to 200 Ksi. It is not
recommended to use through-hardening above 400HB
because of inconsistent performance of the gear in service.
Typically,case hardening is used when there is a desire to
achieve a surface hardness above 400HB.
• The hardness measurement for the allowable bending stress
number is to be taken at the root of the teeth because that is
where the highest bending stress occurs. The allowable
contact stress number is related to the surface hardness on the
face of the gear teeth where the mating teeth experience high
contact stresses.
• When selecting a material for gears, the designer must specify
one that can be hardened to the desired hardness.
• Chapter 2 for heat treatment techniques.
• Appendices 3 and 4 for representative data.
• For the higher hardness (above 250HB) , a medium-carbonalloy steel with good hardenability is desirable. Examples are
AISI 3140,4140,4340,6150,and 8650.
Case-hardened steels
• Flame hardening,induction hardening,carburizing, and
nitriding are processes used to produce a high hardness in
the surface layer of gear teeth. See Fig.2-9.
• In addition to Grade 1 and Grade 2 as described earlier,
case-hardened steel gears can be produced to Grade 3 which
requires an even higher standard of control of the metallurgy
and processing of the material.
Flame- and induction-hardened gear teeth
• Local heating of the surface of the gear teeth by :
(1) High-temperature gas flames
(2) Electrical induction coils.
Normally, medium-carbon-alloy steels (approximately 0.40% to
0.60% carbon) are specified that can be hardened to a resulting
hardness of HRC 50-54. Appendix 3 and 4 list some suitable
• Carburizing produces surface hardnesses in the
range of 55 to 64 HRC. It results in some of the
highest strengths in common use for gears. Special
carburizing steels are listed in Appendix 5. Figure
9-10 shows the AGMA recommendation for the
thickness of the case for carburized gear teeth. The
effective case depth us defined as the depth from
the surface to the point where the hardness has
reached 50 HRC.
• Nitriding produces a very hard but very thin case.
It is specified for applications in which loads are
smooth and well known. Nitriding should be
avoided when over-loading or shock can be
experienced, because the case is not sufficiently
strong or well supported to resist such loads.
Iron and bronze gear materials
• Cast irons. Two types of iron used for gears are gray cast
iron and ductile( also called modular) iron. Table 9-14
gives the common ASTM grades used, with their
corresponding allowable bending stress numbers and
contact stress numbers.
• Austempered ductile iron (ADI) is being used in
some important automotive applications.
Four families of bronzes are typically used for gears:
(1) phosphor or tin bronze;
(2) manganese bronze;
(3) Aluminum bronze;
(4) silicon bronze.
most bronzes are cast, but some are available in wrought
corrosion resistance, good wear properties,
and low friction coefficients, are some advantages of bronzes
Plastic gear materials
• Plastics perform well in applications where low weight,
quiet operation,low friction,good corrosion resistance, and
good wear properties are desired.
• Their strengths are significantly lower than those of most
metallic gear materials,plastics are used in relatively
lightly loaded devices.
• Often plastic gears can be molded into final form without
subsequent machining which gives production cost
Some of the plastic materials used for gears are as follows:
polyphenylene sulfide
Polycarbonate polyester
polyester elastomer
Styrene-acrylonitrile (SAN)
Acrylonitrile-butadiene-styrene (ABS)
these and other plastics can be produced in many formulations
and can be filled with a variety of fillers to improve strength,
wear resistance,impact resistance,temperature limit,moldability,
and other properties.
table 9-5
9-8 stresses in gear teeth
• The stress analysis of gear teeth is facilitated by
consideration of the orthogonal force components,Wt and
Wr, produces a bending moment on the gear tooth similar
to that on a cantilever beam. The resulting bending stress is
maximum at the base of the tooth in the fillet that joins the
involute profile to the bottom of the tooth space. Taking
the detailed geometry of the tooth into account.
Lewis equation
• Wilfred Lewis developed the equation for the
stress at the base of the involute profile.
Wt Pd
t 
where Wt  tangential force
(9 - 12)
Pd  diametral pitch of the tooth
F  face width of the tooth
Y  Lewis form factor, which depends on the tooth form,
the pressure angle, the diametral pitch, the number of teeth in the gear,
and the place where Wt acts
The lewis equation is modified for practical design and analysis.
• One important limitation is that it indicates a stress
concentration in the fillet at the root of the tooth as well as
high contact stresses at the mating surface.
• Comparing the actual stress at the root with that predicted
by the lewis equation enables us to determine the stress
concentration factor,Kt, for the fillet area.
Wt Pd K t
t 
(9 - 13)
• Stress concentration factor - Kt,that depend on the form of
the tooth,the shape and size of the fillet at the root of the
tooth, and the point of application of the force on the tooth.
• Lewis form factor-Y, also depends on the tooth geometry.
• The geometry factor-J, Kt and Y are combined into J, the
value of J varies with the location of the point of application
of the force on the tooth because Y and Kt vary.
where J  Y/K t
Figure 9-15 shows graphs giving the values for the geometry
factor for 20° and 25°,full-depth,involute teeth.
the safest value to use is the one for the load applied at the
tip of the tooth.
Using the geometry factor J
Wt Pd
t 
(9 - 14)
Bending stress number,St
The design analysis method used here is based primarily on
AGMA Standard 2001-C95. Values for some of the factors
are not included in the standard,data from other sources are
added. These data illustrate the kinds of conditions that
affect the final design. The designer ultimately has the
responsibility for making appropriate design decisions.
Wt Pd
Ko Ks Km K B Kv
(9 - 15)
where K 0  overload factor for bending strength
St 
K s  size factor for bending strength
K m  load distributi on factor for bending strength
K B  rim thickness factor
K v  dynamic factor for bending strength
Methods fro specifying values for these factors are discussed below
Overload Factor, Ko
• Overload factors consider the probability that load
veriations,vibrations,shock,speed changes,and other
application-specific conditions may result in peak loads
greater than Wt being applied to the gear teeth during
• many industries have established suitable values based on
• For problem solution in this book, we will use the values
shown in Table 9-6.
• An overload factor of 1.00 would be applied for a perfectly
smooth electric motor driving a perfectly smooth generator
through a gear-type speed reducer. Any rougher conditions
call for a value of K0 greater than 1.00.
For power sources,we will use the following:
• Uniform: Electric motor or constant-speed gas turbine;
examples: continuous-duty generator;
• Light shock: water turbine,variable-speed drive;
examples: Fans and low-speed centrifugal pumps, liquid
agitators, variable-duty generators,uniformly loaded
conveyors,rotary positive displacement pumps;
• moderate shock: Multicylinder engine.
examples: high-speed centrifugal pumps, reciprocating
pumps and compressors, heavy-duty conveyors,machine tool
drives,concrete mixers,textile machinery,meat grinders,saws.
• Heavy shock: Rock crushers, punch press drives, pulverizers,
processing mills, tumbling barrels, wood chippers,vibrating
screens, railroad car dumpers.
Size factor, Ks
• The size factor can be taken to be 1.00 for most gears. But
for gears with large-size teeth or larger widths,a value
greater than 1.00 is recommended.
Reference 13,Table 9-7.
Load-distribution factor,Km
• The determination of the load-distribution factor is based
on many variables in the design of the gears themselves as
well as in the shafts,bearings,housings,and the structure in
which the gear drive is installed. Therefore, it is one of the
most difficult factors to specify. Much analytical and
experimental works is continuing on the determination of
values for Km.
If the intensity of loading on all parts of all teeth in
contact at any given time were uniform,the value of
Km would be 1.00. But this is seldom the case.
• Following factor can cause misalignment of the teeth on the
pinion relative to those on the gear:
• 1. Inaccurate gear teeth;
• 2.misalignment of the axes of shafts carrying gears;
• 3. Elastic deformations of the gears, shafts, bearings, housing,
and support structures;
• 4. Clearances between the shafts and the gears, the shafts and
the bearings, or the bearings and the housing;
• 5. Thermal distortions during operation;
• 6. Crowning or end relief of gear teeth.
The designer can minimize the load-distribution
factor by specifying the following:
1. Accurate teeth (a high quality number);
2. Narrow face widths;
3. Gears centered between bearings(straddle mounting);
4. Short shaft spans between bearings;
5. Large shaft diameters (high stiffness)
6. Rigid, stiff housings;
7. High precision and small clearance on all drive
Following equation is used for computing the value
of the load-distribution factor:
K m  1.0  C pf  C ma
(9 - 16)
where C pf  pinion proportion factor (Fig.9 - 16)
C ma  mesh alignment factor (Fig.9 - 17)
In this book,designs are limited to those with face widths of
15 in or less. Wider face widths call for additional factors.
Rim thickness factor,KB
• The basic analysis used to develop the Lewis
equation assumes that the gear tooth behaves as a
cantilever attached to a perfectly rigid support
structure at its base. If the rim of the gear is too
thin, it can deform and cause the point of
maximum stress to shift from the area of the geartooth fillet to a point within the rim.
Fig.9-18 can be used to estimate the influence of rim
thickness. The key geometry parameter is called the
backup ratio,m ,where
m B  t R / ht
t R  rim thickness
h t  whole depth of the gear tooth
For m B  1.2, the rim is sufficient ly strong and stiff to support th e tooth,
and K B  1.0. The K B factor can also be used in the vicinity of a keyseat
where a small thickness of metal occurs between th e top of the keyseat
and the bottom of the tooth sapce.
Dynamic Factor,Kv
• The dynamic factor accounts for the factor that the load is
assumed by a tooth with some degree of impact and that
the actual load subjected to the tooth is higher than the
transmitted load alone. The value of Kv depends on the
accuracy of the tooth profile, the elastic properties of the
tooth, and the speed with which the teeth come into contact.
Figure 9-19 shows a graph of the AGMA-recommended values
for Kv, where the Qv number are the AGMA-quality numbers
referred to earlier in section9-5
• Gears in typical machine design would fall into the classes
represented by curves 5,6,or 7,which are for gears made by
hobbing or shaping with average to good tooling.
• If the teeth are finish-ground or shaved to improve the
accuracy of the tooth profile and spacing, curve 8,9,10,or
11 should be used.
• Under very special conditions where teeth of high
precision are used in applications where there is little
chance of developing external dynamic loads, the shaded
area can be used.
• If the teeth are cut by form milling, factors lower than
those found from curve 5 should be used.
• The quality 5 gears should not be used at pitch line speeds
above 2500ft/min.
• The dynamic factors are approximate.
• For severe applications,especially those operating above
4000ft/min,approaches taking into account the material
properties, the mass and inertia of the gears, and the actual
error in the tooth form should be used to predict the
dynamic load.(reference10,13,14)
9-9 selection of gear material based on bending stress
• For safe operating, it is the designer’s responsibility to
specify a material that has an allowable bending stress
greater than the computed stress due to bending from
Equation (9-15). Recall that in section 9-6,allowable stress
numbers, Sat, were given for a variety of commonly used
gear materials. Then it is necessary that
S t  S at
These data are valid for the following conditions
Temperature less than 250°F;
107 cycles of tooth loading;
Reliability of 99%:less than one failure in 100;
Safety factor of 1.00
Adjusted Allowable Bending Stress Numbers ,Sat
• Designers may be also choose to apply a factor of
safety to the allowable bending stress number to
account for uncertainties in the design analysis,
material characteristics, or manufacturing
tolerances, or to provide an extra measure of
safety in critical applications.
These factors are applied to the value of Sat to produce an adjusted
allowable bending stress number which we will refer to as Sat
S at'  S at YN /( SF  K R )
(9 - 17)
Stress Cycle Factor,YN
• Figure 9-20 allows the determination of the life adjustment
factor,YN,if the teeth if the gear being analyzed are
expected to experience a number of cycles of loading much
different from 107.
• The general type of material is a factor in this chart for the
lower number of cycles. For the higher number of cycles, a
range is indicated by a shaded area.
• General design practice would use the upper line of this
• Critical applications where pitting and tooth wear must be
minimal may use the lower part of the range.
Calculation of the expected number of cycles of
loading can be done using:
N c  (60)( L)( n)( q)
(9 - 18)
where N C  expected number of cycles of loading
L  design life in hours
n  rotational speed of the gear in rpm
q  number of load applicatio ns per revolution
• Design life is,indeed, a design decision based on the
application. As a guideline, we will use a set of data
created for use in bearing design and reported here as Table
• Unless stated otherwise, we will use a design life
L=20,000h as listed for general industrial machines.
• The normal number of load applications per revolution for
any given tooth is typically one.
Reliability Factor,KR
• Table 9-9 gives data that adjust for the design
reliability desired. These data are based on
statistical analyses of failure data.
Factor of Safety,SF
The factor of safety may be used to account for following:
• Uncertainties in the design analysis;
• Uncertainties in material characteristics;
• Uncertainties in manufacturing tolerances
• It may also be used to provide an extra measure of safety
in critical applications.
• No general guidelines are published, and designers must
evaluate the conditions of each application.
• Many of the factors often considered to be a part of a
factor of safety in general design practice have already
been included in the calculations for St and Sat . Therefore,
a modest value for factor of safety should suffice between
1.00 and 1.50.
Procedure for selecting gear materials for bending stress
• The bending stress number from equation(9-15)
must be less than the adjusted allowable bending
stress number from equation(9-17). That is,
S t  S at'
let's equate the expressions for these two values:
Wt Pd
K o K S K m K B K v  S t  S at
(9 - 19)
to use this relationsh ip for material selection, it is convenient to solve for Sat :
S t  Sat
(9 - 20)
We will use this equation for selecting gear materials based on bending
stress. The list bellow summarize the terms included in Equations(9-19)
and (9-20) for your reference.
Wt  tangential force on gear teeth  (63000)P/n
Pd  diametral pitch of the gear
F  face width of the gear
J  geometry factor for bending stress (Fig.9 - 15)
K o  overload factor (Table 9 - 6)
Ks  size factor (Table 9 - 7)
K m  mesh alignment factor  1.0  C pf  C ma ( Fig .9  16,9  17)
K B  rim thickness factor (Fig.9 - 18)
K v  velocity factor (Fig. 9 - 19)
K R  reliabilit y factor (Table 9 - 9)
S F  factor of safety (design decision)
YN  bending strength stress cycle number (Fig.9 - 20)
• Completing the calculation of the value of the left side of
Equation(9-20) gives the required value for the allowable
bending stress number,Sat .
• Consider first whether the material should be steel, cast
iron,bronze,or plastic.
For steel material,the following review should aid your selection
Start by checking Fig.9-8 to see whether a throughhardened steel will give the needed Sat. if so,determine
the required hardness. Then specify a steel material and
is heat treatment by referring to Appendices 3 and 4.
If a higher Sat is needed, see Table 9-3 and Fig.9-12 and
Fig.9-13 for properties of case-hardened steels;
Appendix 5 will aid in the selection of carburized steels.
If flame or induction hardening is planned,specify a
material with a good hardenability .such as AISI 4140 or
4340 or similar medium-carbon-alloy steels.see
Appendix 4.
Refer to Fig.9-10 or 9-11 for recommended case depths
for surface-hardened steels.
9-10 pitting resistance of gear teeth
• Gear teeth must be safe from failure of the teeth by
breakage, and they must also be capable of running for the
desired life without significant pitting of the tooth form.
• Pitting is the phenomenon in which small particles are
removed from the surface of the tooth because of the high
contact forces that are present between mating teeth.
Pitting is actually the fatigue failure of the tooth surface.
• Prolonged operation after pitting begins causes the tooth
form to change dramatically, causing vibration and noise.
• The design process should seek to prevent the start of
3 齿轮胶合
1 齿轮折断
2 齿轮点蚀
图6-5 齿面磨损
4 齿轮塑性变形
• The primary property of the gear tooth that provides
resistance to pitting is the hardness of the tooth surface.
The harder the surface,the higher is the compressive
strength of the material. The force exerted by the driving
tooth on the mating tooth is theoretically applied to the line
of contact between the tooth profiles.
• Actually,because of the elasticity of the material of the
teeth, the force is spread over a small rectangular area. The
resulting stress is highly concentrated around the area of
application if the force and is called a Contact Stress or a
Hertz Stress.
• The development of the equation for the contact stress on
gear teeth is based on the analysis for two cylinders under
a radial load. The radii of the cylinders are taken to be the
radii of curvature of the involute-tooth forms of the mating
teeth at the point of contact. The load on the teeth is the
total normal load, found from
WN  Wt / cos 
(9 - 21)
where WN  load acting normal to the tooth surface
Wt  transmitted load acting tangentia l to the pitch line
  pressure angle
Hertz contact stress on gear teeth
• The resulting Hertz contact stress due to this load is
FD p
c 
 1  v 2p 1  vG2  cos  sin  mG
 Ep
mG  1
where, in addition t o the terms already defined,
D p  pitch diameter of the pinion
v P  Poisson ' s ratio for the pinion material
v G  Poisson ' s ratio for the gear material
E p  mod ules of elasticity for the pinion material
E G  mod ules of elasticity for the gear material
m G  gear ratio  N G / N P
(9 - 22)
Elastic Coefficient
• The second term under the radical is dependent on
the material properties and is given the name
elastic coefficient,Cp. That is
Cp 
 1  v
E p  1  vG2 / EG
(9 - 23)
for commonly used gear materials,
the value of C p can be found in Table 9 - 10.
The denominator of the last term under the radical in Equation(9-22)
is called the geometry factor,I,and is dependent on the tooth
geometry and on the gear ratio. Values can be found in Fig.9-21.
• I factor for only two tooth forms are included in Fig.9-22 ,
and the values are valid only for those forms. Designers
must ensure that I factors for the tooth form actually used
are included in the contact stress analysis.
• Substituting Cp and I into equation(9-22) gives
 C  Cp
FD p I
(9 - 24)
Contact Stress Number
As with the equation for bending stress in gear teeth, several
factors are added to the equation for contact stress as shown
below. The resulting quantity is called the contact stress
number,sc,the contact stress equation that we will use in
problem solutions is shown below
SC  C p
Wt K 0 K s K m k v
FD p I
(9 - 25)
k 0  overload factor; k s  the size factor;
k m  distributi on factor;
k v  the dynamic factor;
those values of factors can be taken to be the same as
the correspond ing values for the bending stress analysis
in the preceding sections.
9-11 selection of gear material based on contact stress
• Because the pitting resulting from the contact stress is a
different failure phenomenon from tooth failure due to
bending, an independent specification for suitable
materials for the pinion and the gear must now be made.a
in general, the designer must specify a material having an
allowable contact stress number,Sac,greater than the
computed contact stress number, Sc,that is ,
S c  S ac
In section 9-6,value for Sac were given for several materials
that are valid for 107 cycles of loading at a reliability of 99% if
the material temperature is under 250°F. For different life
expectancy and reliability,other factors are added:
S c  S ac
( SF )k R
(9 - 26)
Designs in this book are limited to applications where the
operating temperature is less than 250°F and so on
temperature factor is applied. Data for the reduction in
hardness and strength as a function of temperature should be
sought if higher temperature are experienced.
the reliability factor,kR,is the same as that for bending stress;
it is given in Table 9-9. The other factors in equation(9-26)
are discussed below.
Pitting resistance stress cycle factor,ZN
• The term ZN is the pitting resistance stress cycle factor and
accounts for an expected number of contacts different from
107 as was assumed when the data were produced for the
allowable contact stress number.
• Fig.9-22 shows values for ZN where the solid-line curve is for
most steels and the dashed-line curve is for nitrided steels.
The number of cycles of contact is computed from equation
(9-18) and is the same as that used for bending. For the higher
number of cycles, a range is indicated by the shaded area.
General design practice would use the upper line of this range.
Critical applications where pitting and tooth wear must be
minimal may use the lower part of the range.
Factor of safety, SF
• The factor of safety is based on the same conditions as
described for bending, and often the same value would be
used for both bending and pitting resistance.
• If there are different levels of uncertainty, a different value
should be chosen.
• A modest value for factor of safety should suffice between
1.00 and 1.50.
Hardness ratio factor,CH
• Good gear design practice calls for making the pinion teeth
harder than the gear teeth so that the gear teeth are
smoothed and work-hardened during operation. This
increases the gear capacity with regard to pitting resistance
and is accounted for by the factor CH. Fig.9-23 shows data
for CH for through-hardened gears that depend on the ratio
of the hardness of the pinion and the hardness of the
gear,expressed as the Brinell hardness number, and on the
gear ratio where mG=NG/Np. Use the given curves for
hardness ratios between 1.2 and 1.7. For hardness ratio
under 1.2,use CH=1.00. For hardness ratios over 1.7, use
the value of CH for 1.7, as no substantial additional
improvement is gained.
• CH is applied for the calculations for only the gear,not the
Procedure for selecting gear materials for pitting resistance
• We can refer to the value on the right side of equation(9-26)
as the modified allowable contact stress number, as it
accounts for nonstandard conditions under which the gears
operate that are different from those assumed when the
area for Sac were determined as reported in section 9-6.
• Transposing the modifying factors from equation(9-26) to
the left side of the equation gives
K R ( SF )
sC  s ac
(9 - 27)
Procedure for determining the required properties of most
metallic materials
Solve for the contact stress number,Sc from equations (9-25)
and (9-26) using the same factors as those used for bending
stress number;
Use the value for KR from the bending stress analysis, or
evaluate it from Table 9-9;
Use fig.9-22 to find ZN;
Use fig.9-23 to find that value for CH for through-hardened
gears, or use Fig.9-24 for a surface-hardened pinion and a
through-hardened gear.
5。 Specify a safety factor,typically between 1.00
and 1.50,considering the degree of uncertainty in
material property data, the gear precision,the
severity of the application. Or the danger
presented by the application.
6。 Compute Sac from equation (9-27);
7。 Refer to data in section 9-7, “gear materials” to
select a suitable material. You should consider
first whether the material should be steel, cast
iron,bronze, or plastic. Then consult the related
tables of data.
For steel materials, the following review should aid your selection
Start by checking fig.9-9 to see whether a through-hardened
steel will give the needed Sac. If so,determine the required
hardness. Then specify a steel material and its heat treatment
by referring to Appendices 3 and 4.
If a higher Sac is needed, see Table9-3 for properties of casehardened steels.
Appendix 5 will aid in the selection of carburized steels.
If flame or induction hardening is planned,specify a material
with a good hardenability. Such as AISI 4140 or 4340 or
similar medium-carbon-alloy steels. See appendix 4.
Refer to fig.9-10 or 9-11 for recommended case depths for
surface-hardened steels.
Design of spur gears
• In design involving gear drives,normally the required speeds
of rotation of the pinion and the gear and the amount of
power that the drive must transmit are known. these factors
are determined from the application. Also, the environment
and operating conditions to which the drive will be subjected
must be understood. It is particularly important to know the
type of driving device and the driven machine. In order to
judge the proper value for the application factor.
•The designer must decide the type of gears to use; the
arrangement of the the gears on their shafts; the materials of
gears,including their heat treatment; and the geometry of the
gears: numbers of teeth, diametral pitch,pitch diameters,tooth
form,face width,and quality numbers.
•Design procedure accounts for the bending fatigue strength
of the gear teeth and the pitting resistance,called surface
•There is no one best solution to a gear design problem.
Several good designs are possible.
Design objectives
The resulting drive should
Be compact and small
Operate smoothly and quietly
Have long life
Be low in cost
Be easy to manufacture
Be compatible with the other elements in the machine,such
as bearings,shafts,the housing, the driver, and the driven
Design procedure
Procedure for determining a safe and long-lasting gear drive
1. A geometry that satisfies the required velocity ratio and application
limitations,such as center distance and physical size, is proposed.
2. A tentative choice of the type of material to be used (steel, cast iron)
is made.
3. A trial diametral pitch is chosen. because of its strong effect on
strength,pitting resistance, and geometry,the choice of diametral
pitch is critical. The remaining parts of the procedure are directed
at confirming that a reasonable design can be completed. If not,
the results will help you choose the next trial pitch.
4. The loads,face width, and design factors are determined.
5. The bending stress on the pinion teeth is computed. If a
reasonable value results, the procedure continues. Otherwise, a
new pitch or revised geometry is selected.
6. The contact stress on the surface of the teeth is computed
and the required material properties determined to ensure
against pitting.
7. The final specifications of the materials for the pinion and
the gear are made to satisfy the requirements of both strength
and pitting resistance.
Design guideline
• Figures 9-25 shows a graph of power capacity of a pair of
steel gears versus the speed of rotation of the pinion, with
several values of diametral pitch shown. The hardness for the
pinion and the gear used for these curves are in the middle
range of possible values for steel ( approximately HB300).
This should give you some feel for where to start.
• The curves assume a uniform load (K0=1.0) and good
alignment (Km=1.0). If your application has higher values for
either factor, the entire set of curves is shifted downward.
That is , you should choose a lower value for diametral pitch
than indicated on the graph.
• the face width, F, can be specified once the diametral pitch is
chosen. Although a wide range of face widths is possible, the
following limits are used for general machine drive gears:
Nominal face width
8 / Pd  F  16 / Pd
Norm min al value of F  12/Pd
(9 - 28)
Also, the face width normally is not greater than the pitch
diameter of the pinion. An upper limit is placed on the face width
to minimize problems with alignment. A very wide face width
increases the chance for less than full face loading of the teeth.
When the face width is less than the lower limit of equation(9-28),
it is probable that a more compact design can be achieved with a
different pitch.
The following relationships should help you determine what
changes in your design assumptions you should make after
the first set of calculations to achieve a more optimum design:
• Decreasing the numerical value of the diametral pitch results in
larger teeth and generally lower stresses. Also, the lower value
of the pitch usually means a larger face width,which decreases
stress and increases surface durability.
• Increasing the diameter of the pinion decreases the transmitted
load,generally lowers the stresses, and improves the surface
• Increasing the face width lowers the stresses and improves the
surface durability,but to a generally lesser extent than either the
pitch or the pitch diameter changes discussed previously.
• Gears with more and smaller teeth tend to run more
smoothly and quietly than gears with fewer and larger teeth.
• Standard values of diametral pitch should be used for ease
of manufacture and lower cost (see table 8-2).
• Using high-alloy steels with high surface hardness results
in the most compact system,but the cost is higher.
• using very accurate gears (with ground or shaved teeth)
results in lower dynamic loads and consequently lower
stresses and improved surface durability,but the cost is
• The number of teeth in the pinion should generally be as
small as possible to make the system compact. But the
possibility of interference is greater with fewer teeth.
9-13 Gear design for the metric module system
• Section 8-4, “Gear nomenclature and gear-tooth features,”
described the metric module system of gearing and its
relation to the diametral pitch system. As the design
process was being developed in section9-6 through 911,the data for stress analysis and surface durability
analysis were taken from charts using U.S. Customary
units (in,lb,hp,ft/min ,and Ksi.) Data for the metric module
system were also available in the charts in units of
millimeters (mm), newtons (N),kilowatts,meters per
second(m/s), and megapascals(Mpa). But to use the SI data,
we must modify some of the formulas.
Example problem 9-6 uses SI units. The procedure
will be virtually the same as the used to design with
U.S. Customary units. Those formulas that are
converted to SI units are identified.
9-14 computer aided spur gear design and analysis
• This section presents one approach to assisting the
gear designer with the many calculations and
judgments that must be made to produce an
acceptable design. The spreadsheet shown in
figure 9-26 facilitates the completion of a
prospective design for a pair of gears in a few
minutes by an experienced designer. You must
have studied all of the material in Chapters 8 and 9
in order to understand the data needed in the
spreadsheet and to use it effectively.
Discussion of the use of the spur gear design spreadsheet
1.describing the application;
2. Initial input data;
3. Number of gear tooth;
4.computed data;
5.secondary input data;
6.Factors in design analysis;
7.Alignment factor;
•8.Overload,size, and rim thickness factors;
•9. Dynamic factor;
•10. Service factor;
•11. Hardness ratio factor;
•12. Reliability factor;
•13. Stress cycle factors;
•14. Stress analysis for bending and pitting resistance;
•15. Specification of the materials and heat treatment .
9-15 use of the spur gear design spreadsheet
• Note that the most significant effects on stresses are produced by
the following variable:
• Diametral pitch, Pd;
• Pitch diameter of the pinion, Dp;
• Face width, F;
• Quality number,Qv.
9-16 power transmitting capacity
• When similar materials are used for both the
pinion and the gear, it is likely that the pinion will
be critical for bending stress. But the most critical
condition is usually pitting resistance. The
following relationship can be used to compute the
power-transmitting capacity. In this analysis, it is
assumed that the operating temperature of the
gears and their lubricants is 250°F and that gear
are produced with the appropriate surface finish.
• We start with equation(9-19) in which the computed
bending stress number is compared with the modified
allowable bending stress number for the gear:
Wr Pd
K 0 K s K m K B K v  S t  S at
( SF ) K R
(9 - 19)
but solving for W t gives
s at YN FJ
Wt 
( SF ) K R K 0 K s K m K B K v K d
(9 - 29)
it was shown in equation (9 - 6) that
Wt  (126000)( P ) /( n P D p )
then substituti ng into equation (9 - 29) gives
s at YN FJ
(126000)(P )
n p Dp
( SF ) K R K 0 K s K m K B K v Pd
(9 - 6)
9-17 practical considerations for gears and interfaces with other elements
• It is important to consider the design of the entire gear system when
designing the gears because they must work in harmony with the
other elements in the system.
• Our discussion so far has been concerned primarily with the gear
teeth, including the tooth form, pitch, face width, material selection,
and heat treatment. Also to be considered is the type of gear blank.
Figures 8-2 and 8-4 show several styles of blanks. Smaller gears
and lightly loaded gears are typically made in the plain style.
•Gears with pitch diameters of approximately 5.0 in through 8.0
in are frequently made with thinned webs between the rim and
the hub for lightening, with some having holes bored in the
webs for additional lightening. Larger gears, typically with pitch
diameters greater than 8.0 in, are made from cast blanks with
spokes between the rim and the hub.
•In many precision special machines and gear systems produced
in large quantities, the gears are machined integral with the shaft
carrying the gears. This eliminates some of the problems
associated with mounting and location of the gears, but it may
complicate the machine operations.
• In general machine design, gears are usually mounted on
separate shafts, with the torque transmitted from the shaft
to the gear through the key. This setup provides a positive
means of transmitting the torque while permitting easy
assembly and disassembly. The axial location of the gear
must be provided by another means, such as a shoulder on
the shaft, a retaining ring, or a spacer.
• Other consideration include the forces exerted on the shaft
and bearings that are due to the action of the gears. These
subjects are discussed in section 9-3. The housing design
must provide adequate support for the bearings and
protection of the interior components. Normally, it must
also provide a means of lubricating the gears.
• The action of spur gear teeth is a combination of rolling and sliding,
and because of the high local forces exerted at the gear faces, adequate
lubrication is critical to smoothness of operation and gear life.
• A continuous supply of oil at the pitch line is desirable for most gears
unless they are lightly loaded or operate only intermittently.
• In splash-type lubrication , one of the gears in a pair dips into an oil
supply sump and faces of the case; then it flows down, in a controlled
fashion, onto the pitch line. Simultaneously, the oil can be directed to
the bearings that support the shafts. One difficulty with the splash type
of lubrication is that the oil is churned; at high gear speeds, excessive
heat can be generated, and foaming can occur.
• A positive oil circulation system is used for high-speed and highcapacity systems. A separate pump draws the oil from the sump and
delivers it at a controlled rate to the meshing teeth.
• The primary functions of gear lubricants are to reduce frication at the
mesh and to keep operating temperatures at acceptable levels.
• It is essential that a continuous film of lubricant be
maintained between the mating tooth surfaces of highly
loaded gears and that there be a sufficient flow rate and
total quantity of oil to maintain cool temperatures.
• Heat is generated by the meshing gear teeth, by the
bearings, and by the bearings, and by the churning of the
oil. This heat must be dissipated from the oil to the case or
to some other external heat-exchange device in order to
keep the oil itself below 160F ( approximately 70°C).
Above this temperature, the lubricating ability of the oil, as
indicated by its viscosity, is severely decreased. Also,
chemical changes can be produced in the oil,decreasing is
F-- 法氏度,Fahrenheit(F); °C—摄氏度,
Commercially available gear-type speed reducers
• By studying the design of commercially available gear-type speed
reducers, you should get a better feel for these design details and the
relationships among the component parts: the gears, the shafts, the
bearings ,the housing, the means of providing lubrication, and the
coupling to the driving and driven machines.
• Figure 9-32 .9-33,9-34, shows a double-reduction spur gear speed
reducer with an electric motor rigidly attached. Such a unit is often
called a gear motor.
• The planetary reducer in figure9-35 has quite a different design to
accommodate the placement of the sun, planet, and ring gears. Figure
9-36 shows the eight-speed transmission from a large farm tractor and
illustrates the high degree of complexity that may be involved in the
design of transmissions.
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