| name | spring-design |
| description | Complete spring design — helical compression/tension/torsion, leaf springs, Belleville washers. Wahl factor, fatigue, buckling, natural frequency, clash allowance, material selection. |
| metadata | {"priority":7,"promptSignals":{"phrases":["spring","spring design","spring rate","compression spring","tension spring","torsion spring","Belleville","leaf spring","Wahl"],"minScore":4}} |
Spring Design — Complete Skill
Helical Compression Spring
Geometry
- d = wire diameter
- D = mean coil diameter (OD - d)
- C = spring index = D/d (optimal: 4-12, prefer 6-9)
- N_a = number of active coils
- L_f = free length, L_s = solid length = N_t·d
- N_t = total coils, N_t = N_a + 2 (ground ends), N_a + 1 (closed ends)
Wahl Correction Factor
K_w = (4C-1)/(4C-4) + 0.615/C (curvature + direct shear)
K_B = (4C+2)/(4C-3) (Bergsträsser — slightly different, same order)
Shear Stress
τ = K_w · 8FD/(πd³) = K_w · 8FC/(πd²)
τ_max occurs at inner fiber
Spring Rate
k = Gd⁴/(8D³N_a) = Gd/(8C³N_a)
Modulus of rigidity G:
- Steel (hard drawn): G = 79.3 GPa (11.5×10⁶ psi)
- Stainless 302: G = 68.9 GPa
- Phosphor bronze: G = 41.4 GPa
- Beryllium copper: G = 48.3 GPa
Deflection
δ = 8FD³N_a/(Gd⁴) = FN_a/(kG/d) = F/k
Buckling Check
L_f/D > 5.26 (parallel ends, fixed-free) → buckling possible
L_f/D > 2.63 (parallel ends, fixed-fixed)
Use stability chart: if operating length ratio below critical line, OK
Clash Allowance
δ_clash = L_f - L_s (total available deflection)
Minimum clash: 10-15% of L_f (prevents solid contact under overload)
Fatigue Analysis (Zimmerli data)
σ_a = K_w·8F_a·D/(πd³), σ_m = K_w·8F_m·D/(πd³)
Peened spring endurance: S_sa = 241 MPa, S_su = 0.67Sut
Unpeened: S_sa = 179 MPa
Goodman line: S_sa/S_su + σ_a/S_su + σ_m/S_su = 1 → 1/n = σ_a/S_sa + σ_m/S_su
Set (Solid Height Interference)
τ_solid = K_w·8·F_s·D/(πd³) must be < Ssy = 0.65Sut (no permanent set)
Or: presetting/preset — intentionally set to remove residual stress
Natural Frequency
f_n = (d/4πN_aD²)·√(G/2ρ) Hz (must be > 13× forcing frequency)
Helical Tension Spring
- Initial tension F_i = πd³τ_i/(8D) (approximately 0.5Ssy·d³π/8D)
- F = F_i + k·δ
- Hook stress: σ_hook = 16FD/πd³ · (4C₁-1)/(4C₁-4) + 4F/πd² (bending + direct)
C₁ = 2r₁/d where r₁ = hook mean radius
Torsion Spring
- Bending governs (not torsion)
- σ = K_i · 32M/(πd³) where K_i = (4C²-C-1)/(4C(C-1)) — inner fiber
- Spring rate: k = Ed⁴/(64DN_a) [N·m/rad or lbf·in/rad]
- N_a changes with deflection: N_a' = N_a + θ/360° (θ in degrees)
Belleville Washers (Disc Springs)
- Load: P = 4E/(1-ν²) · t³/K₁D₀² · [(h/t)(h/t - δ/t)(h/t - δ/2t) + (δ/t)³]
K₁ = constant depending on ratio D₀/Di
- Stacking: series increases deflection, parallel increases load
- Typical h/t ratio: 1.41 (maximum energy storage), 0.5-2.0 range
Leaf Springs
- Multi-leaf: k = Ebt³n/(6L³) (n = number of leaves, simply supported)
- σ_max = 3PL/(nbt²)
- Master leaf and helper leaves: consider interleaf friction
- Eye design: stress concentration at clamp
Material Selection
| Material | Sut (MPa) | Max Temp | Application |
|---|
| Hard-drawn wire (A227) | 1310-1760 | 120°C | General purpose |
| Music wire (A228) | 1550-2070 | 120°C | High stress, fatigue |
| Chrome-vanadium (A232) | 1310-1650 | 220°C | Elevated temp, fatigue |
| Chrome-silicon (A401) | 1790-2000 | 250°C | High temp, shock |
| 302 Stainless (A313) | 1100-1500 | 260°C | Corrosion resistance |
| Phosphor bronze | 650-900 | 95°C | Electrical, non-magnetic |
Output
Provide: d selected, D selected, N_a, L_f, k, τ_max, n_fatigue (if cyclic), natural frequency, buckling status.
