AGMA 2001 / ISO 6336 Intermediate Machine Elements & Gearing

Involute Spur Gear Mesh Dynamics & Backlash

Track tooth contact along the line of action, verify contact ratio CR ≥ 1.4, and check sliding velocity.

Constructs exact involute gear profiles from base circles at standard pressure angles (20°). Computes pitch circle diameters, pitch line velocity, contact ratio, and tooth bending stress.

Governing Physical Law & Equations

r_b = r \cos \alpha, \quad CR = \frac{\sqrt{r_{a1}^2 - r_{b1}^2} + \sqrt{r_{a2}^2 - r_{b2}^2} - C\sin\alpha}{p_b}

Base circle radius and contact ratio formula along the line of action.

Law: Fundamental Law of Conjugate Gearing & AGMA 2001 | Standard Reference: AGMA 2001-D04 / ISO 6336 / DIN 3962 (Involute Spur Gear Rating & Geometry)

Adjustable System Parameters

Parameter	Nominal Value	Dynamic Range	Physical Role
Gear Module m (m)	4 mm	2 to 10 mm	Metric tooth size pitch module
Pinion Teeth z1 (z_1)	18	14 to 32	Number of teeth on driving input pinion
Gear Teeth z2 (z_2)	48	24 to 72	Number of teeth on driven output gear
Pressure Angle α (α)	20 °	14.5 to 25 °	Standard involute tooth pressure angle
Pinion Input Speed (N_1)	600 RPM	100 to 1800 RPM	Rotational speed of driving pinion

Analytical Proof & Derivation

Involute tooth profiles obey the Fundamental Law of Gearing: the common normal to tooth profiles at contact always passes through the fixed pitch point P. Pitch diameters are d₁ = m z₁, d₂ = m z₂. Base radii are r_b1 = (d₁/2) cos(α), r_b2 = (d₂/2) cos(α). Path of contact length g_α = √(r_a1² - r_b1²) + √(r_a2² - r_b2²) - C sin(α). Contact ratio CR = g_α / (π m cos(α)). Under AGMA 2001, CR must exceed 1.20 to guarantee smooth tooth handover.

Verification Benchmark

AGMA 2001 benchmark: module m = 4 mm, z₁ = 18, z₂ = 48, α = 20°. Pitch diameters d₁ = 72 mm, d₂ = 192 mm; center distance C = 132 mm. Path of contact g_α = 18.27 mm; base pitch p_b = 11.808 mm. Contact ratio CR = 18.27 / 11.808 = 1.547 (AGMA compliant).

Field Engineering Insights

AGMA 2001 specifies minimum contact ratio CR ≥ 1.4 for industrial gear drives to prevent impact noise and tooth scuffing. AGMA bending stress formula evaluates tooth root bending fatigue life exceeding 10⁷ cycles.