Mechanical Engineering Design · Part 3 — Design of Mechanical Elements

Spur and Helical Gears

Strength-based gear design: the Lewis bending equation and its evolution into the AGMA bending and contact-stress methodology, the full set of modifying factors (geometry, dynamic, load-distribution, size, and more), the allowable-stress numbers, the bending and wear safety factors, and a complete analysis-and-design workflow.

Learning objectives

Distinguish nominal transmitted load, local tooth-root stress, contact stress, and material strength.
Select bending and pitting equations and identify the assumptions behind Lewis and AGMA models.
Apply supplied geometry, dynamic, overload, size, distribution, life, reliability, and temperature factors consistently.
Calculate separate bending and contact margins and identify the governing failure mode.
Iterate module, face width, quality, material, heat treatment, and support geometry toward a feasible mesh.
Document the standard edition and provenance of every rating factor, allowable stress number, and material property.

Physical meaning

The tangential load is a nominal force inferred from torque. Tooth-root bending stress and flank contact stress are local responses produced by that load, while allowable stress numbers represent fatigue strength under stated reference conditions. A factor of safety is meaningful only when stress and strength use the same units, failure mode, life basis, reliability basis, temperature, and standard edition.

Problem-solving strategy

Freeze the duty cycle, ratio, life, reliability, temperature, lubrication, and overload definition before choosing geometry.
Draw the mesh and shaft free bodies, obtain \(W^t=2T/d_P\), and keep nominal force separate from local stresses.
Select one internally consistent AGMA unit system and record the source and applicability of every empirical factor.
Evaluate pinion and gear bending and contact margins independently; never treat static yield as a substitute for fatigue rating.
Change one design variable at a time, recalculate dependent factors, and close with deflection, alignment, thermal, manufacturing, and test checks.

14–1The Lewis Bending Equation

The Tooth as a Cantilever Beam

Wilfred Lewis (1892) modeled a gear tooth as a cantilever beam loaded by the tangential force \(W^t\) at its tip, with the bending stress concentrated at the root fillet. Introducing the Lewis form factor \(Y\) (a function of tooth geometry and number of teeth) folds the beam geometry into one term. The result is the foundational gear bending-stress equation, on which the entire AGMA method is built.

Lewis Bending Equation

\[ \sigma = \frac{W^t\,P}{F\,Y} \qquad (\text{or } \sigma = \frac{W^t}{F\,m\,Y}\ \text{in SI}) \]Eq. (14–7)

EXAMPLE 1Lewis screening stress for a spur tooth

A preliminary metric spur-gear layout carries a nominal tangential load \(W^t=2.20\ \text{kN}\) with module \(m=3.00\ \text{mm}\), face width \(F=30.0\ \text{mm}\), and supplied Lewis form factor \(Y=0.330\). Calculate the nominal Lewis root-bending stress.

Recognition strategy

The requested result is an early geometry screen, and all Lewis variables are supplied. It is not an AGMA rating because dynamic, distribution, rim, life, reliability, and material-strength data are absent.

Given information

\(W^t=2200\ \text{N}\), \(m=3.00\ \text{mm}\), \(F=30.0\ \text{mm}\), and \(Y=0.330\).
The supplied dimensionless \(Y\) applies to the stated tooth form and load position.
Static nominal load, full face contact, cantilever action, and no local factor corrections are assumed.

Model and interpretation

Isolate one tooth as a cantilever loaded tangentially. The SI Lewis relation \(\sigma=W^t/(FmY)\) estimates nominal root stress; it neither predicts local dynamic stress nor supplies fatigue strength.

Step-by-step solution

1. Standardize units. Convert \(2.20\ \text{kN}\) to \(2200 \text{N}\), retaining millimetres so the resulting stress is \(\text{N/mm}^2=\text{MPa}\).

2. Form the resisting term. \(FmY=(30.0)(3.00)(0.330) =29.70\ \text{mm}^2\), an effective geometric area rather than the physical tooth section.

3. Calculate nominal stress. \(\sigma=2200/29.70=74.07 \text{N/mm}^2=74.07\ \text{MPa}\), keeping guard digits for the audit.

4. Classify the result. Rounded to the given precision, \(\sigma=74.1\ \text{MPa}\); no safety factor can be computed without a compatible corrected strength.

Verification

Units: Newtons divided by \(\text{mm}\times\text{mm}\) yield \(\text{N/mm}^2\), which is exactly megapascals, while the Lewis factor is dimensionless.

Signs and direction: The reported value is tensile-stress magnitude at the loaded root flank; the opposite flank and reversed torque swap the tensile side.

Plausibility: An effective area near \(30 \text{mm}^2\) carrying \(2200\ \text{N}\) should produce about \(73\ \text{MPa}\), supporting the computed magnitude.

Final result: \(\boxed{\sigma_{\mathrm{Lewis}}=74.1 \text{MPa}\ \text{nominal root-bending stress}}\)

Engineering interpretation

This number compares candidate tooth sizes under identical assumptions, but it is not allowable load or strength. Final design requires current standard factors, material and heat-treatment certification, tooth quality, life and reliability targets, alignment, rim support, and a separate contact-fatigue assessment.

The Velocity (Dynamic) Factor

Real gears carry dynamic loads above the nominal \(W^t\) from tooth-impact, errors, and inertia, increasing with pitch-line velocity. Early practice multiplied by a velocity factor \(K_v\); AGMA later redefined \(K_v\) as the reciprocal of the older usage. Lewis also recognized the root stress concentration, captured later by a fatigue factor \(K_f\) in the refined approach.

14–2Surface Durability

Contact (Hertzian) Stress and Pitting

Besides bending, gear teeth fail by surface fatigue (pitting) from repeated Hertzian contact pressure. Modeling the meshing teeth as two cylinders in contact gives the surface (contact) stress \(\sigma_c\), which depends on the transmitted load, the contacting radii, the face width, and an elastic coefficient \(C_p\) bundling both materials' elastic constants. This is the second of the two independent failure modes every gear must survive.

Hertzian Contact Stress (cylinder model)

\[ \sigma_c = -C_p\sqrt{\frac{W^t}{F\,\cos\phi}\left(\frac{1}{r_1} + \frac{1}{r_2}\right)} \]Eq. (14–14)

14–3AGMA Stress Equations

The Two Governing Stresses

The AGMA method computes two stresses, each the Lewis/Hertz base modified by a chain of factors. The bending stress \(\sigma\) carries the overload, dynamic, size, load-distribution, and rim factors over the geometry factor \(J\). The contact stress \(\sigma_c\) carries the same load factors plus the elastic coefficient and surface-condition factor over the geometry factor \(I\). Both are then compared with corresponding allowable strengths.

AGMA Bending Stress (U.S. units)

\[ \sigma = W^t\,K_o\,K_v\,K_s\,\frac{P_d}{F}\,\frac{K_m\,K_B}{J} \]Eq. (14–15)

AGMA Contact Stress (U.S. units)

\[ \sigma_c = C_p\sqrt{W^t\,K_o\,K_v\,K_s\,\frac{K_m}{d_P\,F}\,\frac{C_f}{I}} \]Eq. (14–16)

14–4AGMA Strength Equations

Allowable Stress Numbers

AGMA tabulates an allowable bending stress number \(S_t\) and an allowable contact stress number \(S_c\), both rising with hardness and graded by material quality (grade 1, grade 2). For through-hardened steels these are linear in Brinell hardness, e.g. \(S_t = 77.3 H_B + 12\,800\) psi (grade 1) and \(S_c = 322 H_B + 29\,100\) psi (grade 1). The raw numbers are at \(10^7\) cycles and 0.99 reliability and are then adjusted by the life and reliability factors.

Allowable Numbers vs. Brinell Hardness (grade 1, through-hardened)

\[ S_t = 77.3\,H_B + 12\,800\ \text{psi} \qquad\qquad S_c = 322\,H_B + 29\,100\ \text{psi} \]Eqs. (14–17, 14–18)

14–5Geometry Factors I and J (Z_I and Y_J)

Folding Geometry into a Single Number

The bending geometry factor \(J\) refines the Lewis form factor by including the root-fillet stress concentration and the load sharing of the contact ratio. The pitting geometry factor \(I\) combines the radii of curvature at the contact point and the load sharing into the contact-stress base. Both are read from AGMA charts/tables as functions of tooth numbers and pressure angle (and helix angle for helical gears).

Pitting Geometry Factor (external gears)

\[ I = \frac{\cos\phi_t\,\sin\phi_t}{2\,m_N}\,\frac{m_G}{m_G + 1} \]Eq. (14–23)

14–6The Elastic Coefficient Cp (Z_E)

Material Pairing in Contact

The elastic coefficient \(C_p\) packages the elastic moduli and Poisson's ratios of the meshing pinion and gear into the contact-stress equation. For two steel gears it is about 2300 √psi. It is the gear-mesh analog of the contact elastic constant from the surface-fatigue work of Chapter 6.

Elastic Coefficient

\[ C_p = \left[\pi\left(\frac{1 - \nu_P^2}{E_P} + \frac{1 - \nu_G^2}{E_G}\right)\right]^{-1/2} \]Eq. (14–13)

14–7Dynamic Factor Kv

Accounting for Internal Dynamic Load

The dynamic factor \(K_v\) accounts for the load amplification from tooth inaccuracies and resonance at speed. It rises with pitch-line velocity \(V\) and falls with manufacturing quality, indexed by a transmission accuracy number \(Q_v\) (higher = more precise). AGMA gives \(K_v\) as an empirical function of \(V\) and \(Q_v\); a maximum velocity caps the validity of the curve.

Dynamic Factor (AGMA empirical form)

\[ K_v = \left(\frac{A + \sqrt{V}}{A}\right)^{B}, \quad A = 50 + 56(1 - B), \quad B = 0.25(12 - Q_v)^{2/3} \]Eqs. (14–27)–(14–29)

14–8Overload Factor Ko

External Shock

The overload factor \(K_o\) accounts for externally applied load variation beyond the nominal — shock from the driven machine or prime mover. It is selected from a table by the character of the power source (uniform, light shock, medium shock) and the driven machine, much like the application factor in bearing selection.

14–9Surface Condition Factor Cf (Z_R)

Finish and Residual Stress

The surface-condition factor \(C_f\) (used only in the contact-stress equation) accounts for surface finish, residual stress, and work-hardening effects on pitting resistance. AGMA recommends \(C_f = 1\) for normal finishes and a value above 1 where a detrimental surface condition is known.

14–10Size Factor Ks

Size Effect on Strength

The size factor \(K_s\) reflects the reduced strength of larger teeth (the same size effect seen in fatigue). It depends on tooth size (module/diametral pitch), face width, and material. AGMA suggests \(K_s = 1\) unless a size effect is established, with an empirical expression (≈ a function of \(F\sqrt{Y}/P\)) available for larger teeth.

14–11Load-Distribution Factor Km (K_H)

Uneven Load Across the Face

The load-distribution factor \(K_m\) accounts for non-uniform load across the face width from misalignment, shaft deflection, and mounting. It is built from a pinion-proportion factor, a mesh-alignment factor, and a lead-correction term, and grows with face width. Keeping \(F\) within ~3–5 times the circular pitch, and mounting gears near bearings, keeps \(K_m\) low — this factor often dominates the stress calculation.

Load-Distribution Factor

\[ K_m = 1 + C_{mc}\,(C_{pf}\,C_{pm} + C_{ma}\,C_e) \]Eq. (14–30)

14–12Hardness-Ratio Factor CH (Z_W)

Pinion-Gear Hardness Mismatch

The hardness-ratio factor \(C_H\) (contact-stress, applied to the gear only) credits the gear's pitting resistance when the harder pinion work-hardens the softer gear surface during running-in. It depends on the pinion-to-gear hardness ratio and the gear ratio, and is \(\ge 1\). It recognizes the common practice of making the pinion (which sees more cycles) harder than the gear.

14–13Stress-Cycle Factors YN and ZN

Life Other Than 10⁷ Cycles

The allowable stress numbers are based on \(10^7\) cycles. The stress-cycle factors \(Y_N\) (bending) and \(Z_N\) (contact) scale the strength for other lives — above 1 for shorter lives, below 1 for longer. Both equal 1 at \(10^7\) cycles, and their equations change slope on either side of that point. The pinion and gear may have different cycle counts (the pinion turns \(m_G\) times as often), so each gets its own factor.

Stress-Cycle Factors (representative, > 10⁷ cycles)

\[ Y_N = 1.3558\,N^{-0.0178} \qquad\qquad Z_N = 1.4488\,N^{-0.023} \]Figs. (14–14, 14–15)

14–14Reliability Factor KR (Y_Z)

Adjusting from 0.99 Reliability

The allowable numbers correspond to 0.99 reliability (1% failure). The reliability factor \(K_R\) adjusts the strength for other reliability goals — below 1 for lower reliability, above 1 for higher. AGMA gives it as a logarithmic function of the failure probability, e.g. \(K_R = 0.658 - 0.0759\ln(1 - R)\) for \(0.90 \le R \le 0.999\).

Reliability Factor

\[ K_R = 0.658 - 0.0759\,\ln(1 - R) \quad (0.90 \le R \le 0.99) \]Eq. (14–38)

14–15Temperature Factor KT (Y_θ)

Elevated-Temperature Operation

The temperature factor \(K_T = 1\) for lubricant/gear-blank temperatures up to about 250 °F (120 °C). Above that, \(K_T > 1\) reduces the effective strength, computed from the absolute-temperature ratio. Most gearsets run cool enough that \(K_T = 1\).

14–16Rim-Thickness Factor KB

Thin-Rim Gears

When a gear is made with a thin rim over a web or spokes (rather than a solid blank), the rim can bend and add to the root stress. The rim-thickness factor \(K_B\) (bending only) increases the stress for low backup ratios \(m_B = t_R/h_t\) (rim thickness over tooth height). For \(m_B \ge 1.2\), \(K_B = 1\); below that it climbs per a logarithmic relation.

Rim-Thickness Factor (m_B < 1.2)

\[ K_B = 1.6\,\ln\frac{2.242}{m_B}, \qquad m_B = \frac{t_R}{h_t} \]Eq. (14–40)

14–17Safety Factors SF and SH

Bending and Wear Margins

Two safety factors close the method. The bending safety factor \(S_F\) is the fully corrected bending strength over the bending stress. The wear (contact) safety factor \(S_H\) is the corrected contact strength over the contact stress. Because contact stress varies with the square root of load, comparing the two threats requires squaring \(S_H\) (or comparing \(S_F\) with \(S_H^2\)) to judge whether bending or pitting is the governing failure mode.

Bending Safety Factor

\[ S_F = \frac{S_t\,Y_N/(K_T\,K_R)}{\sigma} \]Eq. (14–41)

Wear (Contact) Safety Factor

\[ S_H = \frac{S_c\,Z_N\,C_H/(K_T\,K_R)}{\sigma_c} \]Eq. (14–42)

Comparing the Threats

Since \(\sigma_c \propto \sqrt{W^t}\) while \(\sigma \propto W^t\), the load margins are compared as \(S_F\) vs. \(S_H^2\) (spur) — the smaller indicates the governing failure mode. This lets the designer see at a glance whether a mesh is bending-limited or wear-limited.

14–18Analysis

The Analysis Road Map

To analyze an existing mesh: compute \(W^t\), \(V\), and all the factors (\(K_o, K_v, K_s, K_m, K_B, C_f, C_p, I, J\)); evaluate the bending stress \(\sigma\) and contact stress \(\sigma_c\) for both pinion and gear; evaluate the allowable strengths with their life, reliability, and temperature adjustments; then compute \(S_F\) and \(S_H\) for each. The smallest margin (with \(S_H\) squared for comparison) identifies the weakest link and the governing failure mode.

EXAMPLE 2 Auditable bending margin with supplied factors

For a metric spur pinion, calculate bending stress and bending safety factor using \(W^t=3.20\ \text{kN}\), \(m=2.50\ \text{mm}\), \(F=35.0\ \text{mm}\), \(J=0.340\), \(K_o=1.25\), \(K_v=1.18\), \(K_s=1.00\), \(K_m=1.22\), and \(K_B=1.00\). Supplied rating data are \(S_t=310\ \text{MPa}\), \(Y_N=0.950\), \(K_T=1.00\), and \(K_R=1.10\).

Recognition strategy

All factors for the metric AGMA-form bending equation and corrected bending strength are explicitly supplied. The task is analysis of one fixed geometry, not material selection or a static-yield check.

Given information

Load and geometry are \(W^t=3200\ \text{N}\), \(m=2.50\ \text{mm}\), \(F=35.0\ \text{mm}\), and \(J=0.340\).
Stress factors are supplied as \(K_o=1.25\), \(K_v=1.18\), \(K_s=1.00\), \(K_m=1.22\), \(K_B=1.00\).
Strength-side values are supplied rating inputs, whose standard edition and material basis must be verified before release.

Model and interpretation

Use \(\sigma=W^tK_oK_vK_sK_mK_B/(FmJ)\) and corrected strength \(S_{\mathrm{allow}}=S_tY_N/(K_TK_R)\). Stress multipliers belong only on the load side; reliability and life corrections belong on the strength side.

Step-by-step solution

1. Combine load-side factors. \(K_oK_vK_sK_mK_B =(1.25)(1.18)(1.00)(1.22)(1.00)=1.7995\), without premature rounding.

2. Evaluate bending stress. \(\sigma=3200(1.7995)/ [(35.0)(2.50)(0.340)]=193.56\ \text{MPa}\).

3. Correct the strength. \(S_{\mathrm{allow}}=310(0.950)/ [(1.00)(1.10)]=267.73\ \text{MPa}\), which is fatigue-rating strength, not induced stress.

4. Calculate the margin. \(S_F=267.73/193.56=1.383\); report \(1.38\) while retaining the inputs and factor provenance for traceability.

Verification

Units: The numerator is newtons and \(Fm\) is square millimetres; all factors are dimensionless, so stress is \(\text{N/mm}^2=\text{MPa}\) and \(S_F\) is dimensionless.

Signs and direction: Rating uses positive stress magnitude; tooth-side tension reverses with torque direction, but neither that reversal nor compression permits a negative safety factor.

Plausibility: The unfactored Lewis-type stress is \(3200/29.75=107.6\ \text{MPa}\); multiplication by \(1.7995\) gives \(193.6\ \text{MPa}\), independently matching the result.

Final result: \(\boxed{\sigma=193.6\ \text{MPa}, S_{\mathrm{allow}}=267.7\ \text{MPa},\ S_F=1.38}\)

Engineering interpretation

The calculated fatigue margin is above unity but cannot be declared adequate without a specified design target and a separate pitting calculation. Confirm factor values against the applicable standard edition, actual quality, alignment, rim geometry, duty spectrum, heat treatment, and inspection records.

14–19Design of a Gear Mesh

From Requirements to a Mesh

Design inverts the analysis. With the power, speed, ratio, and life/reliability goals given, the designer chooses tentative tooth numbers (interference-free), a diametral pitch, a face width (typically \(3p \le F \le 5p\)), and a material/hardness, then computes \(S_F\) and \(S_H\) and iterates — usually adjusting hardness or face width — until both meet the target with the desired margin. A decision set (tooth numbers, pitch, face width, quality, material) is varied systematically; spreadsheets make the trade-off study practical.

EXAMPLE 3 Face-width iteration governed by two failure modes

A fully factored spur-mesh analysis at face width \(F_0=24.0 \text{mm}\) gives bending stress \(260\ \text{MPa}\), contact stress \(980\ \text{MPa}\), corrected bending strength \(360\ \text{MPa}\), and corrected contact strength \(1170\ \text{MPa}\). Holding all supplied factors constant for this local iteration, find the minimum width for load margins \(S_F\ge1.50\) and \(S_H^2\ge1.50\), then audit a manufacturable \(28.0\ \text{mm}\) choice.

Recognition strategy

With all other factors frozen, bending stress scales as \(F^{-1}\) and contact stress as \(F^{-1/2}\). Therefore bending margin and squared contact margin both scale linearly with width, enabling a transparent first iteration.

Given information

Baseline: \(F_0=24.0\ \text{mm}\), \(\sigma_0=260\ \text{MPa}\), and \(\sigma_{c0}=980\ \text{MPa}\).
Corrected fatigue strengths are \(360\ \text{MPa}\) in bending and \(1170\ \text{MPa}\) in contact.
The factor-freezing assumption is only a local sensitivity model; a final analysis must update width-dependent \(K_m\), \(K_s\), and geometry.

Model and interpretation

Use \(S_F=S_b/\sigma\), \(S_H=S_c/\sigma_c\), \(\sigma=\sigma_0F_0/F\), and \(\sigma_c=\sigma_{c0}\sqrt{F_0/F}\). Compare equal load margins \(S_F\) and \(S_H^2\), not raw stresses.

Step-by-step solution

1. Compute baseline margins. \(S_{F0}=360/260=1.3846\); \(S_{H0}=1170/980=1.1939\), so \(S_{H0}^2=1.4255\).

2. Size for bending. Since \(S_F\propto F\), \(F_b=24.0(1.50/1.3846)=26.00\ \text{mm}\).

3. Size for contact. Since \(S_H^2\propto F\), \(F_c=24.0(1.50/1.4255)=25.25\ \text{mm}\); bending governs the frozen-factor iteration.

4. Audit the selected width. At \(28.0\ \text{mm}\), \(\sigma=260(24/28)=222.9\ \text{MPa}\), \(S_F=1.615\); \(\sigma_c=980\sqrt{24/28}=907.3\ \text{MPa}\), and \(S_H^2=(1170/907.3)^2=1.663\).

Verification

Units: Width ratios and both safety margins are dimensionless; multiplying the baseline stresses by dimensionless scaling factors correctly retains megapascals.

Signs and direction: Bending and contact rating equations use positive cyclic-stress magnitudes; changing torque direction changes loaded flanks but not these magnitude-based margins.

Plausibility: Increasing width by \(16.7\%\) should lower bending stress by \(14.3\%\) and contact stress by about \(7.4\%\); the computed \(222.9\) and \(907.3\ \text{MPa}\) follow those trends.

Final result: \(\boxed{F_{\min}=26.00\ \text{mm} \ \text{(bending governs)};\ F=28.0\ \text{mm}\Rightarrow S_F=1.615,\ S_H^2=1.663}\)

Engineering interpretation

The rounded width passes only the deliberately frozen-factor model. Wider faces can worsen load distribution, so the next iteration must update \(K_m\), shaft and housing deflection, lead correction, contact pattern, heat flow, process capability, weight, and cost using current standard and vendor data.

Chapter recap

Lewis theory is a useful bending screen; full rating adds dynamic, distribution, size, rim, geometry, and service effects.
Bending fatigue and pitting fatigue are independent failure modes and both pinion and gear require separate checks.
Stress is load response, strength is resistance, and each correction factor belongs to a documented side of that comparison.
Design is iterative because module, width, hardness, quality, alignment, temperature, life, reliability, cost, and manufacturing capability interact.

Before moving on: be able to derive \(W^t\) from torque, select consistent bending and contact equations, audit factor provenance and units, calculate \(S_F\) and \(S_H\), identify the governing load margin, and state which assumptions must be revisited after a geometry change.