Chapter 14site.iugaza.edu.ps/mhaiba/files/2013/09/CH-14-Spur-and-Helical... · The Lewis Bending Equation Assume that max stress in a gear tooth occurs at point a 12/25/2015 12:27

Chapter 14

Spur and

Helical Gears

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Mohammad Suliman Abuhaiba, Ph.D., PE1

The Lewis Bending Equation

Equation to estimate bending stress in gear teeth

in which tooth form entered into the formulation:

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Mohammad Suliman Abuhaiba, Ph.D., PE

2


Assume that max stress in a gear tooth

occurs at point a

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3


Only bending of tooth is considered

Compression due to radial component of

force is neglected

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4


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Table 14–2

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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8


Fatigue stress-concentration factor Kf by

Mitchiner & Mabie

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l & t = layout - Fig. 14.1

f = pressure angle

rf = fillet radius

b = dedendum

d = pitch diameter

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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11

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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12

Table 6–3

Example 14–212/25/2015 12:27 PM


13

kc = kd = ke = 1

Example 14–212/25/2015 12:27 PM


14

If a material exhibited a Goodman failure locus,

Gerber fatigue locus gives mean values of

kf = 1.66

Example 14–212/25/2015 12:27 PM


15

Fig. A–15–6

Kt = 1.68: Fig. 6–20, q = 0.62 ; Eq. (6–32)

Example 14–212/25/2015 12:27 PM


16

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 due to presence of foreign

material

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Eq. (3–74):

contact stress

between two

cylinders

pmax = largest

surface pressure

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18

Figure 3–38


For gears, replace F by Wt / cos φ, d by 2r, and l

by face width F

Replacing pmax by σC , 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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20


With this simplification, and the addition of a

velocity factor Kv, Eq. (14–11) can be written as

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21

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

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Example 14–3

F = 1.5 in, Kv = 1.52

Surface endurance strength of cast iron for 108

cycles:

Table A–24: HB = 262 for ASTM No. 50 cast iron

factor of safety = SC /σC

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24

14–3 AGMA Stress Equations

Two fundamental stress equations are

used in the AGMA methodology:

1. For bending stress

2. For contact stress

In AGMA terminology, 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)

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14–3 AGMA Stress Equations - Bending

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

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 Tables 14–3 &

14–4

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Figure 14–2: Allowable

bending stress number for

through-hardened steels

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30

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 EquationsFigure 14–4: Allowable bending stress numbers for nitriding steel gears

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14–4 AGMA Strength EquationsFigure 14–4: Allowable bending stress numbers for

nitriding steel gears

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33

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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14–4 AGMA Strength EquationsTable 14–4: Repeatedly Applied Bending Strength St for Iron &

Bronze Gears at 107 Cycles & 0.99 Reliability for Steel Gears

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35

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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38

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 ObtainedSource: 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 GearsSource: ANSI/AGMA 2001-D04.


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Table 14–7: Repeatedly Applied Contact Strength Sc at 107

Cycles and 0.99 Reliability for Iron and Bronze Gears


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When two-way (reversed) loading

occurs, AGMA recommends using 70 %

of St values.


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43

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 same purpose

in a more involved manner

Face-contact ratio mF

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)

Equation for J for spur and helical gears is

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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Factor Kf in Eq. (14–20): stress-correction

factor by AGMA.

based on a formula deduced from a photo-elastic

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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50

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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Modifying factor can be applied to

J factor when other than 75 teeth

are used in the mating element


Surface-Strength Geometry Factor I (ZI)

Pitting-resistance geometry factor by AGMA

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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 factor = 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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Effects that produce transmission error are:

Inaccuracies produced in generation of tooth

profile

Vibration of tooth during meshing due to tooth

stiffness

Magnitude of pitch-line velocity

Dynamic unbalance of rotating members

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Effects that produce transmission error are:

Wear & permanent deformation of contacting

portions of teeth

Gear shaft misalignment and linear & angular

deflection of 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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Equations for dynamic factor are based on Qv

numbers:

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Maximum velocity, representing the end point of

the Qv curve, is given by

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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 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 = used only in pitting resistance Eq. 14–16

It depends on

Surface finish as affected by cutting, shaving,

lapping, grinding, shot peening

Residual stress

Plastic effects (work hardening)

AGMA specifies a value of Cf greater than unity

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14–10 Size Factor Ks

Size factor reflects non-uniformity of material

properties due to size and 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)

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.

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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

Pinion generally has a smaller number of teeth

than gear and consequently is subjected to

more cycles of contact stress.

If both pinion & gear are through-hardened,

then a uniform surface strength can be

obtained by making pinion harder than 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 surface strengths for this

effect.

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Figure 14–12

Hardness-ratio

factor CH (through-

hardened steel)

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When surface-hardened pinions with 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 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.

Load cycle factors YN & ZN modify gear

strength for lives other than 107 cycles.

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Values for these factors: Figs. 14–14 & 14–15.

For life goals slightly higher than 107 cycles,

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

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

Caution is required when comparing SF with SH

in an analysis in order to ascertain nature and

severity of 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 &

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, f = 20°,, np = 1800 rpm, Pin = 4 hp, NG = 52, P = 10

teeth/in, F = 1.5 in, Qv = 6, straddle-mounted gears.

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

v = 0.30, JP = 0.30, JG = 0.40, E = 30 Mpsi, smooth loading. pinion life

= 108 cycles, R = 0.90, YN = 1.3558N−0.0178, ZN = 1.4488N−0.023,

uncrowned tooth profile, commercial enclosed gear unit

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Documents

Chapter 14site.iugaza.edu.ps/mhaiba/files/2013/09/CH-14-Spur-and-Helical... · The Lewis Bending Equation Assume that max stress in a gear tooth occurs at point a 12/25/2015 12:27