| name | shaft-design |
| description | Complete shaft design — combined bending+torsion, DE-Goodman/ASME-Elliptic criteria, critical speed, deflection, keyway stress concentration, bearing seat sizing. Shigley Ch.6-7. |
| metadata | {"priority":9,"promptSignals":{"phrases":["shaft","shaft design","torque transmission","critical speed","shaft diameter","power transmission shaft"],"minScore":4}} |
Shaft Design — Complete Skill
Step 1: Load Analysis
- Identify all torques (T) — from gears, pulleys, couplings
- Identify all transverse loads (F) — gear forces, pulley tensions
- Build free-body diagram in both planes (horizontal + vertical)
- Compute M_H and M_V at each critical section
- Resultant moment: M = √(M_H² + M_V²)
Step 2: Gear/Pulley Forces
Spur gear: W_t = 33000·HP/V (Imperial), W_t = T/r (SI)
W_r = W_t·tan(φ) (φ = pressure angle, typically 20°)
W_a = 0 (no axial for spur)
Helical gear: W_t = T/r, W_r = W_t·tan(φ_n)/cos(ψ), W_a = W_t·tan(ψ)
Belt drive: T = (F₁-F₂)·r; F₁/F₂ = e^(μθ)
Net radial force on shaft: F = F₁ + F₂ (for V-belt: multiply by 0.9)
Chain: Similar to belt but no slip; F = T/r (tight side only, slack = 0 assumed)
Step 3: Critical Section
Locate where M and T are simultaneously large — usually at gear/bearing junction.
Check: shoulders, keyways, press-fit locations (stress concentration sites).
Step 4: Stress Concentration Factors
| Feature | Kt (bending) | Kts (torsion) |
|---|
| Shoulder fillet r/d=0.02 | 2.7 | 2.2 |
| Shoulder fillet r/d=0.1 | 1.7 | 1.5 |
| Keyway (end milled) | 2.14 | 3.0 |
| Press fit | 2.0-3.0 | — |
| Transverse hole | 2.0-2.4 | 1.5-2.0 |
Fatigue factors: Kf = 1 + q(Kt-1), Kfs = 1 + qs(Kts-1)
Step 5: DE-Goodman Criterion (recommended)
Fully reversed bending + steady torsion (most common shaft loading):
σ_a = Kf · 32M / (πd³)
σ_m = 0 (bending is fully reversed)
τ_a = 0
τ_m = Kfs · 16T / (πd³)
Von Mises:
σ'_a = √(σ_a² + 3τ_a²) = Kf · 32M / (πd³)
σ'_m = √(σ_m² + 3τ_m²) = Kfs · 16T√3 / (πd³)
Goodman safety factor:
1/n = σ'_a/Se + σ'_m/Sut
Required diameter (DE-Goodman):
d = [16n/π · (√(4(KfM)²+3(KfsT)²)/Se + √(4(KfM)²+3(KfsT)²)/Sut·(terms))]^(1/3)
Simplified for fully reversed bending + mean torsion:
d³ = 16n/π · [4(KfMa)²+3(KfsT)²]^0.5/Se + [4(KfMm)²+3(KfsT)²]^0.5/Sut
Step 6: ASME-Elliptic Criterion (alternative)
1/n² = (σ'_a/Se)² + (σ'_m/Sy)²
Step 7: Static Check (first cycle yield)
von Mises stress: σ_von = √(σ² + 3τ²)
σ_max = 32M/πd³ + P/A (add axial if present)
τ_max = 16T/πd³ + VQ/Ib (add transverse shear if significant)
Check: σ_von < Sy (usually not governing if fatigue check passes)
Step 8: Critical Speed (Rayleigh-Ritz)
Single mass: n_c = π/2 · √(g/δ) rpm, δ = static deflection under own weight + loads
Multi-mass (Dunkerley's equation):
1/n_c² = 1/n₁² + 1/n₂² + ... + 1/n_shaft²
Where n_i = critical speed if only mass i were present.
Rule: Operating speed < 0.75·n_c (subcritical) OR > 1.4·n_c (supercritical — pass through quickly)
Shaft alone (simply supported):
n_c = π²/L² · √(EIg/w) rpm (w = weight per unit length)
Step 9: Deflection Check
At bearing locations: δ < 0.001in/ft·L (bearing misalignment limit)
At gears: δ < 0.005" (spur), slope < 0.0005 rad (helical)
At seals: slope < 0.001 rad
Step 10: Keyway Design
Key width w = d/4, height h = d/6 (standard proportions)
Key length L: shear of key: τ = 2T/(dLw) < Ssy
Bearing on key/hub: σ = 4T/(dLh) < 0.9Sy
Minimum Shaft Diameter Selection (AGMA 9002)
d_min from: torque capacity, bending capacity, critical speed, stiffness — take the maximum.
Output
Provide: d (required), d_selected (next standard size), n_fatigue, n_yield, n_c, governing failure mode.
