Spur and Helical Gears - الصفحات...

Chapter 14

Spur and

Helical Gears

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Dr. Mohammad Suliman Abuhaiba, PE 1

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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Dr. Mohammad Suliman Abuhaiba, PE

2


Fig. 14–1b: assume that max stress in a

gear tooth occurs at point a

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Only bending of tooth is considered

Compression due to radial component of

force is neglected

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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)


Dynamic Effects

Barth velocity factor (English units)

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Dynamic Effects

Barth velocity factor (SI units)

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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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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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Eq. (3–74):

contact stress

between two

cylinders

pmax = largest

surface pressure

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Figure 3–38


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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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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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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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


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


St = 0.7255HB + 153.63 Mpa


St = 0.7255HB + 201.91 Mpa


Table 14–3: Repeatedly Applied Bending Strength St at 107

Cycles & 0.99 Reliability for Steel Gears

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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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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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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.


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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


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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


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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


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Figure 14–5: Contact-fatigue strength Sc at 107 cycles and

0.99 reliability for through-hardened steel gears


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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.


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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.


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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.


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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.


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Cycles and 0.99 Reliability for Iron and Bronze Gears


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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.


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When two-way (reversed) loading

occurs, AGMA recommends using 70 %

of St values.


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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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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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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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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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Figure 13–15


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


Figure 14–7: Helical-gear

geometry factors J’

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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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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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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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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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The following equations for the dynamic factor

are based on these Qv numbers:

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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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The load-distribution factor is given by

for values of F/(10d) < 0.05, F/(10d) = 0.05 is used

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Figure 14–10: Definition of distances S and S1 used in

evaluating Cpm, Eq. (14–33).

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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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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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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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Figure 14–12

Hardness-ratio

factor CH

(through-

hardened steel).

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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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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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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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Figure 14–14: Repeatedly applied bending

strength stress-cycle factor YN

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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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14–18 Analysis

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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


14–18 Analysis

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Figure 14–18: Roadmap of gear wear equations based on AGMA


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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