Mechanical Design

Gear Sizing & Bending Stress

Sizes an external spur or helical gear from tooth count and pitch, then estimates root bending stress using the Lewis equation with the AGMA dynamic (velocity) factor. Mating-gear profile shift and downloadable tooth geometry are on the way — see the note at the bottom of the page.

I.

Gear Sizing

Pitch diameter and standard full-depth tooth proportions

Inputs

Shared with the bending-stress section below

Results

Pitch diameter

—

mm

—in

Outside diameter

—

mm

—in

Root diameter

—

mm

—in

Base circle diameter

—

mm

—in

Circular pitch (transverse)

—

mm

—in

Whole depth

—

mm

—in

Transverse pressure angle

—

degrees

Equivalent pitch

—

—

Lead (helical only)

—

mm

—in

Sizing equations (normal system, standard full depth)

mt = mn / cos ψ     d = N·mt
tan φt = tan φn / cos ψ
da = d + 2mn     df = d − 2.5mn     db = d·cos φt
mn = normal module mt = transverse module ψ = helix angle (0 for spur) φn, φt = normal / transverse pressure angle Addendum a = 1.000·mn, dedendum b = 1.250·mn
II.

Bending Stress

Lewis equation with the AGMA dynamic (velocity) factor Kv

Inputs

rpm
Enter a material allowable to get a safety factor

Results

Tangential load Wt

—

N

—lbf

Pitch line velocity

—

ft/min

—m/s

Lewis form factor Y

—

—

AGMA dynamic factor Kv

—

dimensionless

—

Bending stress σ

—

psi

—ksi
—MPa

Safety factor

—

vs. allowable stress

Lewis equation with AGMA dynamic factor

σ = Wt · Kv F · mt · Y
Kv = (A + √Vt) A B    B = 0.25(12−Qv)2/3    A = 50 + 56(1−B)
Wt = tangential load = 2T/d Vt = pitch line velocity, ft/min Y = Lewis form factor (from Nv = N/cos³ψ) Valid for Qv = 5–11 (ANSI/AGMA 2001-D04)
Lewis form factor source Y is interpolated from the classic AGMA/Lewis full-depth tooth-form-factor table (14.5° and 20° systems, N = 10 through rack). For helical gears the virtual tooth count Nv = N/cos³ψ is used to enter the same spur-gear table — the standard practical substitute for a true helical geometry factor.
What this does — and doesn't — cover This is Lewis bending stress with only the AGMA dynamic factor Kv applied. It does not include overload (Ko), load distribution (Km), size (Ks), or rim thickness (KB) factors, nor surface (pitting/contact) stress. Treat results as a first-pass bending check, not a full AGMA 2001 design verification.
III.

Mating Gear & Profile Shift

Undercut avoidance and operating center distance for a pair

Inputs

Gear 1 is the gear defined in Part I — same module, pressure angle, and helix angle apply to both
Makes gear 2 into gear 1 (swaps N↔N₂ and x₁↔x₂) so Part IV can generate its geometry
The −x₁ button balances x₂ to hold the standard center distance

Results

Min. shift, gear 1 (undercut)

—

x₁ below this undercuts

—

Min. shift, gear 2 (undercut)

—

x₂ below this undercuts

—

Sum of profile shifts

—

x₁ + x₂

Operating pressure angle

—

degrees, transverse

Standard center distance

—

mm

—in

Operating center distance

—

mm

—in

Gear 1 — OD / root dia.

—

mm outside dia.

—mm root dia.
—in outside dia.
—in root dia.

Gear 2 — OD / root dia.

—

mm outside dia.

—mm root dia.
—in outside dia.
—in root dia.

Gear 2 — pitch diameter

—

mm

—in

Undercut avoidance and operating center distance

xmin = h*a − (z·sin²φt) / (2·cos ψ)
inv φwt = inv φt + 2(x₁+x₂)tan φn / (z₁+z₂)     inv φ = tan φ − φ
aw = a·cos φt / cos φwt     a = mt(z₁+z₂)/2
h*a = 1.00 (standard full-depth addendum coefficient) Operating pressure angle & center distance solved by inverting inv(φ) numerically da = d + 2mn(1+x)    df = d − 2mn(1.25−x)
Practical shift range Keep x roughly between −0.5 and +0.8 for standard cutting tools — large positive shifts risk a pointed tooth tip (top land thinner than ~0.2× module), and this calculator doesn't check top-land thickness. Verify tip thickness in your CAD model once geometry is generated in Part IV.
Holding standard center distance Setting x₂ = −x₁ (the button above) keeps the sum of shifts at zero, so the operating center distance equals the standard center distance even though the pinion is corrected for undercut. This is the classic "long-and-short addendum" approach.
Coming next on this page: generating the actual involute tooth geometry for gear 1 (using the module, pressure angle, helix angle, and profile shift x₁ from above) with a downloadable DXF file you can extrude in Fusion 360, SolidWorks, or import directly into laser-cutting software.
IV.

Tooth Geometry & DXF Export

Full involute profile for gear 1, ready to extrude or laser-cut

Inputs

Fraction of module; standard full-round rack tip ≈ 0.38. Increased automatically if too small to bridge base and root circles.
Sets the file's $INSUNITS flag so CAD software scales it correctly on import
Geometry is generated for gear 1 (Part I / Part III) at profile shift x₁. This is a 2D transverse-plane profile — for a helical gear, extrude with the twist angle noted below rather than a straight extrude.

Preview

Results

Fillet radius used

—

mm

Tip land thickness

—

× module (0.2 min. recommended)

—

Twist angle (for helical extrude)

—

—

Involute flank and root fillet construction

x(t) = rb(cos t + t·sin t)     y(t) = rb(sin t − t·cos t)
inv(t) = t − arctan(t)     ψ(ρ) = ψr + inv(tr) − inv(t)
Root fillet: circular arc of radius ρ*·mn, tangent to the root circle and to the involute flank's base point
Flank sampled parametrically in roll angle t from the root/base circle to the outside circle Fillet found via tangent-circle intersection (root circle ∩ circle of radius ρ centered on the flank's base point) Full gear built by rotating one tooth N−1 times about the gear center
What the fillet is — and isn't This is a constant-radius circular arc tangent to the root and flank, not a true hob-generated trochoid. It's the standard simplification used by most quick gear-profile tools and is entirely adequate for extruded, printed, or laser-cut parts. If you need the exact hobbed root form for a high-load application, that requires simulating the generating rack's cutting motion — out of scope here.
Helical gears need a twisted extrude The DXF is the 2D transverse profile only. A straight extrude of it gives you a spur-equivalent shape, not a true helical tooth. In Fusion 360, use Extrude → Twist Angle; in SolidWorks, use a Helical Sweep along the axis. The twist angle for a given face width F is 360°·F/Lead, shown above.