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Chapter 14

Spur and Helical Gears

Lecture Slides

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Chapter Outline

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Cantilever Beam Model of Bending Stress in Gear Tooth

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Fig. 14–1

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Lewis Equation

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Lewis Equation

Lewis Form Factor

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Values of Lewis Form Factor Y

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Table 14–2

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Dynamic Effects

Effective load increases as velocity increases
Velocity factor K_v accounts for this
With pitch-line velocity V in feet per minute,

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Dynamic Effects

With pitch-line velocity V in meters per second,

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Lewis Equation

The Lewis equation including velocity factor

U.S. Customary version

Metric version

Acceptable for general estimation of stresses in gear teeth
Forms basis for AGMA method, which is preferred approach

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Example 14–1

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Example 14–1

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Example 14–2

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Example 14–2

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Example 14–2

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Example 14–2

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Example 14–2

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Example 14–2

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Fatigue Stress-Concentration Factor

A photoelastic investigation gives an estimate of fatigue stress-concentration factor as

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Surface Durability

Another failure mode is wear due to contact stress.
Modeling gear tooth mesh with contact stress between two cylinders, From Eq. (3–74),

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Surface Durability

Converting to terms of gear tooth, the surface compressive stress (Hertzian stress) is found.

Critical location is usually at the pitch line, where

Define elastic coefficient from denominator of Eq. (14–11),

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Surface Durability

Incorporating elastic coefficient and velocity factor, the contact stress equation is

Again, this is useful for estimating, and as the basis for the preferred AGMA approach.

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Example 14–3

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Example 14–3

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AGMA Method

The American Gear Manufacturers Association (AGMA) provides a recommended method for gear design.
It includes bending stress and contact stress as two failure modes.
It incorporates modifying factors to account for various situations.
It imbeds much of the detail in tables and figures.

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AGMA Bending Stress

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AGMA Contact Stress

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AGMA Strengths

AGMA uses allowable stress numbers rather than strengths.
We will refer to them as strengths for consistency within the textbook.
The gear strength values are only for use with the AGMA stress values, and should not be compared with other true material strengths.
Representative values of typically available bending strengths are given in Table 14–3 for steel gears and Table 14–4 for iron and bronze gears.
Figs. 14–2, 14–3, and 14–4 are used as indicated in the tables.
Tables assume repeatedly applied loads at 10⁷ cycles and 0.99 reliability.

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Bending Strengths for Steel Gears

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Bending Strengths for Iron and Bronze Gears

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Bending Strengths for Through-hardened Steel Gears

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Fig. 14–2

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Bending Strengths for Nitrided Through-hardened Steel Gears

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Fig. 14–3

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Bending Strengths for Nitriding Steel Gears

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Fig. 14–4

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Allowable Bending Stress

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Allowable Contact Stress

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Nominal Temperature Used in Nitriding and Hardness Obtained

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Table 14–5

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Contact Strength for Steel Gears

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Contact Strength for Iron and Bronze Gears

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Contact Strength for Through-hardened Steel Gears

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Fig. 14–5

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Geometry Factor J (Y_J in metric)

Accounts for shape of tooth in bending stress equation
Includes

A modification of the Lewis form factor Y
Fatigue stress-concentration factor K_f
Tooth load-sharing ratio m_N

AGMA equation for geometry factor is

Values for Y and Z are found in the AGMA standards.
For most common case of spur gear with 20º pressure angle, J can be read directly from Fig. 14–6.
For helical gears with 20º normal pressure angle, use Figs. 14–7 and 14–8.

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Spur-Gear Geometry Factor J

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Fig. 14–6

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Helical-Gear Geometry Factor J

Get J' from Fig. 14–7, which assumes the mating gear has 75 teeth
Get multiplier from Fig. 14–8 for mating gear with other than 75 teeth
Obtain J by applying multiplier to J'

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Fig. 14–7

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Modifying Factor for J

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Fig. 14–8

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Surface Strength Geometry Factor I (Z_I in metric)

Called pitting resistance geometry factor by AGMA

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Elastic Coefficient C_P (Z_E)

Obtained from Eq. (14–13) or from Table 14–8.

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Elastic Coefficient

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Dynamic Factor K_v

Accounts for increased forces with increased speed
Affected by manufacturing quality of gears
A set of quality numbers define tolerances for gears manufactured to a specified accuracy.
Quality numbers 3 to 7 include most commercial-quality gears.
Quality numbers 8 to 12 are of precision quality.
The AGMA transmission accuracy-level number Q_v is basically the same as the quality number.

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Dynamic Factor K_v

Dynamic Factor equation

Or can obtain value directly from Fig. 14–9
Maximum recommended velocity for a given quality number,

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Dynamic Factor K_v

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Fig. 14–9

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Overload Factor K_O

To account for likelihood of increase in nominal tangential load due to particular application.
Recommended values,

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Surface Condition Factor C_f (Z_R)

To account for detrimental surface finish
No values currently given by AGMA
Use value of 1 for normal commercial gears

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Size Factor K_s

Accounts for fatigue size effect, and non-uniformity of material properties for large sizes
AGMA has not established size factors
Use 1 for normal gear sizes
Could apply fatigue size factor method from Ch. 6, where this size factor is the reciprocal of the Marin size factor k_b. Applying known geometry information for the gear tooth,

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Load-Distribution Factor K_m (K_H)

Accounts for non-uniform distribution of load across the line of contact
Depends on mounting and face width
Load-distribution factor is currently only defined for

Face width to pinion pitch diameter ratio F/d ≤ 2
Gears mounted between bearings
Face widths up to 40 in
Contact across the full width of the narrowest member

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Load-Distribution Factor K_m (K_H)

Face load-distribution factor

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Load-Distribution Factor K_m (K_H)

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Fig. 14–10

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Load-Distribution Factor K_m (K_H)

C_ma can be obtained from Eq. (14–34) with Table 14–9

Or can read C_ma directly from Fig. 14–11

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Load-Distribution Factor K_m (K_H)

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Fig. 14–11

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Hardness-Ratio Factor C_H (Z_W)

Since the pinion is subjected to more cycles than the gear, it is often hardened more than the gear.
The hardness-ratio factor accounts for the difference in hardness of the pinion and gear.
C_H is only applied to the gear. That is, C_H = 1 for the pinion.
For the gear,

Eq. (14–36) in graph form is given in Fig. 14–12.

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Hardness-Ratio Factor C_H

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Fig. 14–12

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Hardness-Ratio Factor

If the pinion is surface-hardened to 48 Rockwell C or greater, the softer gear can experience work-hardening during operation. In this case,

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Fig. 14–13

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Stress-Cycle Factors Y_N and Z_N

AGMA strengths are for 10⁷ cycles
Stress-cycle factors account for other design cycles
Fig. 14–14 gives Y_N for bending
Fig. 14–15 gives Z_N for contact stress

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Stress-Cycle Factor Y_N

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Fig. 14–14

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Stress-Cycle Factor Z_N

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Fig. 14–15

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Reliability Factor K_R (Y_Z)

Accounts for statistical distributions of material fatigue failures
Does not account for load variation
Use Table 14–10
Since reliability is highly nonlinear, if interpolation between table values is needed, use the least-squares regression fit,

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Table 14–10

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Temperature Factor K_T (Y_θ)

AGMA has not established values for this factor.
For temperatures up to 250ºF (120ºC), K_T = 1 is acceptable.

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Rim-Thickness Factor K_B

Accounts for bending of rim on a gear that is not solid

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Fig. 14–16

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Safety Factors S_F and S_H

Included as design factors in the strength equations
Can be solved for and used as factor of safety

Or, can set equal to unity, and solve for traditional factor of safety as n = σ_all/σ

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Comparison of Factors of Safety

Bending stress is linear with transmitted load.
Contact stress is not linear with transmitted load
To compare the factors of safety between the different failure modes, to determine which is critical,

Compare S_F with S_H² for linear or helical contact
Compare S_F with S_H³ for spherical contact

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Summary for Bending of Gear Teeth

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Fig. 14–17

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Summary for Surface Wear of Gear Teeth

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Fig. 14–18

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Example 14–4

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Example 14–4

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Example 14–4

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Example 14–4

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Example 14–4

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Example 14–4

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Example 14–4

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Example 14–4

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Example 14–4

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Example 14–4

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Example 14–4

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Example 14–4

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Example 14–5

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Example 14–5

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Example 14–5

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Example 14–5

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Example 14–5

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Example 14–5

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Example 14–5

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Example 14–5

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Comparing Pinion with Gear

Comparing the pinion with the gear can provide insight.
Equating factors of safety from bending equations for pinion and gear, and cancelling all terms that are equivalent for the two, and solving for the gear strength, we get

Substituting in equations for the stress-cycle factor Y_N,

Normally, m_G > 1, and J_G > J_P, so Eq. (14–44) indicates the gear can be less strong than the pinion for the same safety factor.

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Comparing Pinion and Gear

Repeating the same process for contact stress equations,

Neglecting C_H which is near unity,

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