Chapter 14
Spur and
Helical Gears
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The Lewis Bending Equation
An equation for estimating the bending
stress in gear teeth in which the tooth
form entered into the formulation.
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The Lewis Bending Equation
Fig. 14–1b: assume that max stress in a
gear tooth occurs at point a
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The Lewis Bending Equation
Only bending of tooth is considered
Compression due to radial component of
force is neglected
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The Lewis Bending Equation
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Table 14–2
Values of Lewis
Form Factor Y
(fn = 20°, Full-
Depth Teeth,
Diametral Pitch
of Unity in Plane
of Rotation)
The Lewis Bending Equation
Dynamic Effects
Barth velocity factor (English units)
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The Lewis Bending Equation
Dynamic Effects
Barth velocity factor (SI units)
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The Lewis Bending Equation
Dynamic Effects
Introducing velocity factor into Eq. (14–2) gives
The metric version of this equation is
Spur gears: face width F = 3 to 5 times circular
pitch p
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The Lewis Bending Equation
Fatigue stress-concentration factor Kf by
Mitchiner & Mabie
l & t = from layout in Fig. 14–1
f = pressure angle
rf = fillet radius
b = dedendum
d = pitch diameter
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Example 14–1
A stock spur gear is available having a
diametral pitch of 8 teeth/in, a 1.5” face, 16
teeth, and a pressure angle of 20° with full-
depth teeth. The material is AISI 1020 steel in
as rolled condition. Use a design factor of nd =
3 to rate the horsepower output of the gear
corresponding to a speed of 1200 rpm and
moderate applications.
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Example 14–1
Table A–20: Sut = 55 kpsi & Sy = 30 kpsi.
Nd = 3, allowable bending stress = 30/3 = 10 kpsi
pitch diameter = N/P = 16/8 = 2 in
Table 14–2: form factor Y = 0.296 for 16 teeth
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Example 14–2
Estimate the horsepower rating of the gear in the
previous example based on obtaining an infinite
life in bending.
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Table 6–3
Example 14–2 1/2/2015 2:06 PM
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kc = kd = ke = 1
Example 14–2 1/2/2015 2:06 PM
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If a material exhibited a Goodman failure locus,
Gerber fatigue locus gives mean values of
kf = 1.66
Example 14–2 1/2/2015 2:06 PM
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Fig. A–15–6
Kt = 1.68: Fig. 6–20, q = 0.62 ; Eq. (6–32)
Example 14–2 1/2/2015 2:06 PM
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14–2 Surface Durability
Wear: failure of the surfaces of gear teeth
Pitting: a surface fatigue failure due to many
repetitions of high contact stresses
Scoring: a lubrication failure
Abrasion: wear to the presence of foreign
material
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14–2 Surface Durability
Eq. (3–74):
contact stress
between two
cylinders
pmax = largest
surface pressure
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Figure 3–38
14–2 Surface Durability
For gears, replace F by Wt/cos φ, d by 2r, and l
by the face width F
Replacing pmax by σC , the surface compressive
stress is found from the equation
r1 & r2: instantaneous values of radii of
curvature on pinion & gear-tooth profiles at
point of contact.
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14–2 Surface Durability
First evidence of wear occurs near the pitch line
Radii of curvature of tooth profiles at pitch point:
AGMA defines an elastic coefficient Cp
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14–2 Surface Durability
With this simplification, and the addition of a
velocity factor Kv, Eq. (14–11) can be written as
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Example 14–3
The pinion of Examples 14–1 and 14–2 is to be
mated with a 50-tooth gear manufactured of
ASTM No. 50 cast iron. Using the tangential load
of 382 lbf, estimate the factor of safety of the
drive based on the possibility of a surface fatigue
failure.
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Example 14–3
Table A–5: EP = 30 Mpsi, nP = 0.292, EG = 14.5
Mpsi, nG = 0.211
dP = 2 in, dG = 50/8 = 6.25 in
F = 1.5 in, Kv = 1.52
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Example 14–3
The surface endurance strength of cast iron can
be estimated from for 108
cycles.
Table A–24: HB = 262 for ASTM No. 50 cast
iron.
factor of safety = SC /σC
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14–3 AGMA Stress Equations
Two fundamental stress equations are
used in the AGMA methodology:
1. One for bending stress
2. Another for contact stress
In AGMA terminology, these are called
stress numbers
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14–3 AGMA Stress Equations - Bending
Wt = tangential transmitted load, lbf (N)
Ko = overload factor, Kv = dynamic factor
Ks = size factor, Pd = transverse diametral pitch
F (b) = face width of narrower member, in (mm)
Km (KH) = load-distribution factor
KB = rim-thickness factor
J (YJ) = geometry factor for bending strength (includes
root fillet stress-concentration factor Kf )
mt = transverse metric module
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14–3 AGMA Stress Equations
The fundamental equation for contact stress
Cp (ZE) = elastic coefficient, √lbf/in2 (√N/mm2)
Cf (ZR) = surface condition factor
dP (dw1) = pitch diameter of pinion, in (mm)
I (ZI) = geometry factor for pitting resistance
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14–4 AGMA Strength Equations
Uppercase letter S = strength
Lowercase Greek letters σ and τ = stress
Gear strength = allowable stress numbers as
used by AGMA
Values for gear bending strength, St = Figs.
14–2, 14–3, & 14–4, and in Tables 14–3 &
14–4
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14–4 AGMA Strength Equations
Figure 14–2: Allowable
bending stress number for
through-hardened steels
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St = 0.533HB + 88.3 MPa, grade 1
St = 0.703HB + 113 MPa , grade 2
14–4 AGMA Strength Equations
Figure 14–3: Allowable bending stress number for nitrided
through hardened steel gears (AISI 4140, 4340)
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St = 0.568HB + 83.8 MPa, grade 1
St = 0.749HB + 110 MPa, grade 2
14–4 AGMA Strength Equations Figure 14–4: Allowable bending stress numbers for nitriding steel gears
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14–4 AGMA Strength Equations Figure 14–4: Allowable bending stress numbers for nitriding steel gears
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The SI equations are: 1. Nitralloy grade 1
St = 0.594HB + 87.76 Mpa
2. Nitralloy grade 2
St = 0.784HB + 114.81 Mpa
3. 2.5% chrome, grade 1
St = 0.7255HB + 63.89 Mpa
4. 2.5% chrome, grade 2
St = 0.7255HB + 153.63 Mpa
5. 2.5% chrome, grade 3
St = 0.7255HB + 201.91 Mpa
14–4 AGMA Strength Equations
Table 14–3: Repeatedly Applied Bending Strength St at 107
Cycles & 0.99 Reliability for Steel Gears
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14–4 AGMA Strength Equations
Table 14–3: Repeatedly Applied Bending Strength St at 107
Cycles & 0.99 Reliability for Steel Gears
Notes: See ANSI/AGMA 2001-D04 for references cited in
notes 1–7.
1. Hardness to be equivalent to that at root diameter in the
center of tooth space & face width.
2. See tables 7 through 10 for major metallurgical factors
for each stress grade of steel gears.
3. Steel selected must be compatible with heat treatment
process selected and hardness required.
4. Allowable stress numbers indicated may be used with
the case depths prescribed in 16.1.
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14–4 AGMA Strength Equations
Table 14–3: Repeatedly Applied Bending Strength St at 107 Cycles &
0.99 Reliability for Steel Gears
Notes: See ANSI/AGMA 2001-D04 for references cited in notes 1–7.
5. See figure 12 for type A & type B hardness patterns.
6. If bainite & microcracks are limited to grade 3 levels, 70000 psi
may be used.
7. Overload capacity of nitrided gears is low. Since the shape of
effective S-N curve is flat, sensitivity to shock should be
investigated before proceeding with the design. [7]
8. *Tables 8 & 9 of ANSI/AGMA 2001-D04 are comprehensive
tabulations of the major metallurgical factors affecting St and Sc
of flame-hardened and induction-hardened (Table 8) and
carburized and hardened (Table 9) steel gears.
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14–4 AGMA Strength Equations
Table 14–4: Repeatedly Applied Bending Strength St for Iron and Bronze
Gears at 107 Cycles & 0.99 Reliability for Steel Gears
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Table 14–4: Repeatedly Applied Bending Strength St for Iron and
Bronze Gears at 107 Cycles & 0.99 Reliability for Steel Gears
Notes:
1. See ANSI/AGMA 2004-B89, Gear Materials and Heat Treatment
Manual.
2. Measured hardness to be equivalent to that which would be
measured at the root diameter in the center of the tooth space and
face width.
3. The lower values should be used for general design purposes.
Upper values may be used when: High quality material is used.
Section size and design allow maximum response to heat treatment.
Proper quality control is effected by adequate inspection.
Operating experience justifies their use.
14–4 AGMA Strength Equations
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The equation for the allowable bending stress is
St = allowable bending stress, lbf/in2 (N/mm2)
YN = stress cycle factor for bending stress
KT (Yθ ) = temperature factors
KR (YZ ) = reliability factors
SF = AGMA factor of safety, a stress ratio
14–4 AGMA Strength Equations
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The equation for allowable contact stress σc,all is
Sc = allowable contact stress, lbf/in2 (N/mm2)
ZN = stress cycle life factor
CH (ZW) = hardness ratio factors for pitting resistance
KT (Yθ ) = temperature factors
KR (YZ) = reliability factors
SH = AGMA factor of safety, a stress ratio
14–4 AGMA Strength Equations
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Allowable contact stress, Sc: Fig. 14–5 and Tables
14–5, 14–6, and 14–7.
AGMA allowable stress numbers (strengths) for
bending and contact stress are for
Unidirectional loading
10 million stress cycles
99 % reliability
14–4 AGMA Strength Equations
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Figure 14–5: Contact-fatigue strength Sc at 107 cycles and
0.99 reliability for through-hardened steel gears
14–4 AGMA Strength Equations
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Sc = 2.22HB + 200 MPa, grade 1
Sc = 2.41HB + 237 MPa, grade 2
Table 14–5: Nominal Temperature Used in Nitriding &
Hardnesses Obtained Source: Darle W. Dudley, Handbook of Practical Gear Design, rev. ed., McGraw-Hill, New York, 1984.
14–4 AGMA Strength Equations
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Table 14–6: Repeatedly Applied Contact Strength Sc at 107
Cycles and 0.99 Reliability for Steel Gears Source: ANSI/AGMA 2001-D04.
14–4 AGMA Strength Equations
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Table 14–6: Repeatedly Applied Contact Strength
Sc at 107 Cycles & 0.99 Reliability for Steel Gears
Notes: See ANSI/AGMA 2001-D04 for references
cited in notes 1–5.
1. Hardness to be equivalent to that at the start of
active profile in the center of the face width.
2. See Tables 7 through 10 for major metallurgical
factors for each stress grade of steel gears.
3. Steel selected must be compatible with the heat
treatment process selected and hardness
required.
14–4 AGMA Strength Equations
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Table 14–6: Repeatedly Applied Contact Strength Sc at 107
Cycles and 0.99 Reliability for Steel Gears
Notes: See ANSI/AGMA 2001-D04 for references cited in
notes 1–5.
4. These materials must be annealed or normalized as a
minimum.
5. Allowable stress numbers indicated may be used with
the case depths prescribed in 16.1.
6. Table 9 of ANSI/AGMA 2001-D04 is a comprehensive
tabulation of the major metallurgical factors affecting St
and Sc of carburized and hardened steel gears.
14–4 AGMA Strength Equations
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Table 14–7: Repeatedly Applied Contact Strength Sc at 107
Cycles and 0.99 Reliability for Iron and Bronze Gears
14–4 AGMA Strength Equations
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Table 14–7: Repeatedly Applied Contact Strength Sc 107 Cycles and 0.99
Reliability for Iron and Bronze Gears
Notes:
1. See ANSI/AGMA 2004-B89, Gear Materials and Heat Treatment
Manual.
2. Hardness to be equivalent to that at the start of active profile in the
center of the face width.
3. Lower values should be used for general design purposes.
4. The upper values may be used when:
High-quality material is used
Section size & design allow maximum response to heat treatment.
Proper quality control is effected by adequate inspection.
Operating experience justifies their use.
14–4 AGMA Strength Equations
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When two-way (reversed) loading
occurs, AGMA recommends using 70 %
of St values.
14–4 AGMA Strength Equations
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14–5 Geometry Factors I & J (ZI & YJ)
Y is used in the Lewis equation to introduce the
effect of tooth form into the stress equation.
AGMA factors I & J: accomplish the same
purpose in a more involved manner.
The face-contact ratio mF is defined as
px = axial pitch
F = face width
For spur gears, mF = 0
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14–5 Geometry Factors I & J (ZI & YJ) Bending-Strength Geometry Factor J (YJ)
The equation for J for spur and helical gears is
Form factor Y in Eq. (14–20) is not Lewis factor at all.
Value of Y here is obtained from calculations within
AGMA 908-B89, and is often based on the highest
point of single-tooth contact.
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14–5 Geometry Factors I & J (ZI & YJ) Bending-Strength Geometry Factor J (YJ)
The factor Kf in Eq. (14–20) is called a stress-
correction factor by AGMA.
based on a formula deduced from a photoelastic
investigation of stress concentration in gear teeth.
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14–5 Geometry Factors I & J (ZI & YJ) Bending-Strength Geometry Factor J (YJ)
Load-sharing ratio mN = face width / min total
length of lines of contact
This factor depends on:
Transverse contact ratio mp
Face-contact ratio mF
Effects of any profile modifications, and tooth
deflection
For spur gears, mN = 1.0
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14–5 Geometry Factors I & J (ZI & YJ) Bending-Strength Geometry Factor J (YJ)
For helical gears having a face-contact ratio mF >
2.0, a conservative approximation is given by
pN = normal base pitch
Z = length of line of action in the transverse plane
(distance Lab in Fig. 13–15).
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14–5 Geometry Factors I & J (ZI & YJ) Bending-Strength Geometry Factor J (YJ)
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Figure 13–15
14–5 Geometry Factors I & J (ZI & YJ) Bending-Strength Geometry Factor J (YJ)
Figure 14–6: geometry factor J for spur gears having
a 20° pressure angle and full-depth teeth
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Number of teeth for which factor is desired
14–5 Geometry Factors I & J (ZI & YJ) Bending-Strength Geometry Factor J (YJ)
Figure 14–7: Helical-gear
geometry factors J’
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14–5 Geometry Factors I & J (ZI & YJ) Bending-Strength Geometry Factor J (YJ)
Figure 14–8: J -factor multipliers for use with Fig. 14–7 to find J.
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The modifying factor can be applied to the
J factor when other than 75 teeth are used
in the mating element
14–5 Geometry Factors I & J (ZI & YJ) Bending-Strength Geometry Factor J (YJ) Surface-Strength Geometry Factor I (ZI)
Pitting-resistance geometry factor by AGMA
mN = 1 for spur gears
pN = normal base pitch
Z = length of line of action in the transverse plane (Lab in Fig. 13–15)
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14–5 Geometry Factors I & J (ZI & YJ) Bending-Strength Geometry Factor J (YJ)
rP & rG = pitch radii
rbP & rbG = base-circle radii of pinion & gear
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14–6 The Elastic Coefficient Cp (ZE)
Eq. 14–13
Table 14–8
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14–7 Dynamic Factor Kv
Dynamic factors = account for inaccuracies in
manufacture & meshing of gear teeth in action
Transmission error = departure from uniform
angular velocity of the gear pair
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14–7 Dynamic Factor Kv
Some of the effects that produce transmission error are:
Inaccuracies produced in the generation of tooth profile
Vibration of tooth during meshing due to tooth stiffness
Magnitude of pitch-line velocity
Dynamic unbalance of rotating members
Wear & permanent deformation of contacting portions
of the teeth
Gear shaft misalignment and linear & angular deflection
of the shaft
Tooth friction
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14–7 Dynamic Factor Kv
AGMA has defined a set of quality numbers
These numbers define tolerances for gears of
various sizes manufactured to a specified
accuracy
Quality numbers 3 to 7 = most commercial-
quality gears
Quality numbers 8 to 12 = precision quality
AGMA transmission accuracy level number Qv =
quality number
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14–7 Dynamic Factor Kv
The following equations for the dynamic factor
are based on these Qv numbers:
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14–7 Dynamic Factor Kv
Maximum velocity, representing the end point of
the Qv curve, is given by
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14–7 Dynamic Factor Kv Figure 14–9: Dynamic factor Kv
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14–8 Overload Factor Ko
Intended to make allowance for all externally
applied loads in excess of the nominal
tangential load Wt in a particular application
An extensive list of service factors appears in:
Howard B. Schwerdlin, “Couplings,” Chap. 16
Joseph E. Shigley, Charles R. Mischke, and
Thomas H. Brown, Jr. (eds.), Standard Handbook
of Machine Design, 3rd ed., McGraw-Hill, New
York, 2004.
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Surface Condition Factor Cf (ZR)
Cf or ZR is used only in the pitting resistance
equation, Eq. (14–16).
It depends on
Surface finish as affected by cutting, shaving,
lapping, grinding, shotpeening
Residual stress
Plastic effects (work hardening)
AGMA specifies a value of Cf greater than unity
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14–10 Size Factor Ks
The size factor reflects non-uniformity of
material properties due to size
It depends upon
Tooth size
Diameter of part
Ratio of tooth size to diameter of part
Face width
Area of stress pattern
Ratio of case depth to tooth size
Hardenability and heat treatment
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14–10 Size Factor Ks
Ks is given by
If Ks in the preceding Eq is less than 1, use
Ks = 1.
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14–11 Load-Distribution Factor Km (KH)
The load-distribution factor reflects non-uniform
distribution of load across the line of contact.
The ideal is to locate the gear “mid-span” between
two bearings at the zero slope place when the load is
applied.
The following procedure is applicable to
Net face width to pinion pitch diameter ratio F/d ≤ 2
Gear elements mounted between the bearings
Face widths up to 40 in
Contact, when loaded, across the full width of the
narrowest member
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14–11 Load-Distribution Factor Km (KH)
The load-distribution factor is given by
for values of F/(10d) < 0.05, F/(10d) = 0.05 is used
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14–11 Load-Distribution Factor Km (KH)
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14–11 Load-Distribution Factor Km (KH)
Figure 14–10: Definition of distances S and S1 used in
evaluating Cpm, Eq. (14–33).
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14–11 Load-Distribution Factor Km (KH)
Table 14–9: Empirical Constants A, B, and C for Eq. (14–
34), Face Width F in Inches
Source: ANSI/AGMA 2001-D04
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14–11 Load-Distribution Factor Km (KH)
Figure 14–11: Mesh alignment factor Cma. Curve-fit
equations in Table 14–9
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14–12 Hardness-Ratio Factor CH
The pinion generally has a smaller number of
teeth than the gear and consequently is
subjected to more cycles of contact stress.
If both pinion and gear are through-hardened,
then a uniform surface strength can be
obtained by making the pinion harder than the
gear.
A similar effect can be obtained when a
surface-hardened pinion is mated with a
through hardened gear.
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14–12 Hardness-Ratio Factor CH
Hardness-ratio factor CH is used only for the
gear.
Its purpose is to adjust the surface strengths for
this effect.
Values of CH are obtained from the equation
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14–12 Hardness-Ratio Factor CH
Figure 14–12
Hardness-ratio
factor CH
(through-
hardened steel).
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14–12 Hardness-Ratio Factor CH
When surface-hardened pinions with
hardnesses (Rockwell C48) or harder are run
with through-hardened gears (180–400
Brinell), a work hardening occurs.
CH factor is a function of pinion surface finish fP
and the mating gear hardness. Figure 14–13
displays the relationships:
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14–12 Hardness-Ratio Factor CH
Figure 14–13
Hardness-ratio factor
CH (surface-hardened
steel pinion)
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14–13 Stress-Cycle Factors YN & ZN
AGMA strengths as given in Figs. 14–2
through 14–4, in Tables 14–3 and 14–4
for bending fatigue, and in Fig. 14–5 and
Tables 14–5 and 14–6 for contact-stress
fatigue are based on 107 load cycles
applied.
The purpose of load cycle factors YN & ZN
is to modify gear strength for lives other
than 107 cycles.
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14–13 Stress-Cycle Factors YN & ZN
Values for these factors are given in Figs.
14–14 and 14–15.
For life goals slightly higher than 107
cycles, the mating gear may be
experiencing fewer than 107 cycles and
the equations for (YN)P and (YN)G can be
different.
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14–13 Stress-Cycle Factors YN & ZN
Figure 14–14: Repeatedly applied bending
strength stress-cycle factor YN
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14–13 Stress-Cycle Factors YN & ZN
Figure 14–14: Pitting resistance stress-cycle factor ZN
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14–14 Reliability Factor KR (YZ)
Reliability factor accounts for the effect of
the statistical distributions of material
fatigue failures.
Gear strengths St & Sc are based on a
reliability of 99%
Table 14–10
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14–14 Reliability Factor KR (YZ)
Table 14–10: Reliability
Factors KR (YZ) Source: ANSI/AGMA 2001-D04
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14–15 Temperature Factor KT (Yθ)
For oil or gear-blank temperatures
up to 120°C, use KT = Yθ = 1.0.
For higher temperatures, the factor
should be greater than unity.
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14–16 Rim-Thickness Factor KB
When rim thickness is not sufficient to provide full
support for the tooth root, the location of bending
fatigue failure may be through the gear rim rather
than at the tooth fillet.
Rim-thickness factor KB, adjusts estimated
bending stress for thin-rimmed gear.
It is a function of the backup ratio mB
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14–16 Rim-Thickness Factor KB
tR = rim thickness below the tooth
ht = tooth height
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Figure 14–16
14–17 Safety Factors SF & SH
SF = safety factor guarding against bending
fatigue failure
SH = safety factor guarding against pitting
failure
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14–17 Safety Factors SF & SH
a caution is required when comparing SF with SH
in an analysis in order to ascertain the nature
and severity of the threat to loss of function.
To render SH linear with the transmitted load, Wt
it could have been defined as
with the exponent 2 for linear or helical contact,
or 3 for crowned teeth (spherical contact).
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14–18 Analysis
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Figure 14–17: Roadmap of gear bending equations based on AGMA
standards. (ANSI/AGMA 2001-D04.)
14–18 Analysis
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Figure 14–17: Roadmap of gear bending equations based on AGMA
standards. (ANSI/AGMA 2001-D04.)
14–18 Analysis
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Figure 14–17: Roadmap of gear bending equations based on AGMA
standards. (ANSI/AGMA 2001-D04.)
14–18 Analysis
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Figure 14–18: Roadmap of gear wear
equations based on AGMA standards.
(ANSI/AGMA 2001-D04.)
14–18 Analysis
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Figure 14–18: Roadmap of gear wear equations based on AGMA
standards. (ANSI/AGMA 2001-D04.)
14–18 Analysis
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Figure 14–18: Roadmap of gear wear equations based on AGMA
standards. (ANSI/AGMA 2001-D04.)
Example 14–4 A 17-tooth 20° pressure angle spur pinion rotates at 1800 rpm and
transmits 4 hp to a 52-tooth disk gear. The diametral pitch is 10 teeth/in,
the face width 1.5 in, and the quality standard is No. 6. The gears are
straddle-mounted with bearings immediately adjacent. The pinion is a
grade 1 steel with a hardness of 240 Brinell tooth surface and through-
hardened core. The gear is steel, through-hardened also, grade 1
material, with a Brinell hardness of 200, tooth surface and core.
Poisson’s ratio is 0.30, JP = 0.30,JG = 0.40, and Young’s modulus is
30(106) psi. The loading is smooth because of motor and load. Assume a
pinion life of 108 cycles and a reliability of 0.90, and use YN =
1.3558N−0.0178, ZN = 1.4488N−0.023. The tooth profile is uncrowned. This is
a commercial enclosed gear unit.
a. Find the factor of safety of the gears in bending.
b. Find the factor of safety of the gears in wear.
c. By examining the factors of safety, identify the threat to each gear
and to the mesh.
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Example 14–4 Summary:
Np= 17-tooth, f = 20°, spur pinion, np = 1800 rpm, Pin = 4 hp, NG =
52-diametral pitch =10 teeth/in, F = 1.5 in, Qv = 6, straddle-mounted
gears with bearings immediately adjacent.
pinion material= grade 1 steel, 240 HB tooth surface and through-
hardened core
gear material = steel, through-hardened, grade 1, 200HB, tooth
surface and core
Poisson’s ratio = 0.30, JP = 0.30, JG = 0.40, Young’s modulus = 30(106)
psi, smooth loading. pinion life = 108 cycles, reliability = 0.90, YN =
1.3558N−0.0178, ZN = 1.4488N−0.023, uncrowned tooth profile,
commercial enclosed gear unit.
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