| name | torsion-spring |
| description | Torsion spring design — angular rate, bending stress (not torsion), coil diameter change on deflection, end types, fatigue, Wahl correction. |
| metadata | {"priority":7,"promptSignals":{"phrases":["torsion spring","torsional spring","angular spring rate","spring moment"],"minScore":3}} |
Torsion Spring Design — Complete Skill
Key Distinction
Torsion springs carry bending stress (not torsional), despite the name.
The wire bends as the spring winds/unwinds under applied moment.
Geometry and Parameters
d = wire diameter [mm]
D = mean coil diameter [mm]
C = spring index = D/d
N_a = active coils
θ = angular deflection [rad or degrees]
Body length = N_a × d (coils touching free state typically; or with gaps)
Angular Spring Rate
Rate: k_θ = Ed⁴ / (10.8 DN_a) [N·mm/deg] or
k_θ = Ed⁴ / (64DN_a) × (π/180) [N·mm/deg]
In SI consistent units:
k_θ = Ed⁴ / (64DN_a) [N·mm/rad]
E = elastic modulus (steel: 207 GPa)
Note: E not G — because wire bends, not twists
Moment: M = k_θ × θ [N·mm]
Stress — Wahl Correction for Curvature
Bending stress (inner fiber is critical):
σ = K_i × 32M / (πd³)
Inner surface correction (Wahl):
K_i = (4C²-C-1) / (4C(C-1))
Outer surface correction:
K_o = (4C²+C-1) / (4C(C+1))
K_i > K_o → inner fiber governs for static loading
For fatigue: use K_i (stress concentration is real)
Coil Diameter Change Under Load
As torsion spring deflects, coil diameter changes:
D_wound = D × N_a / (N_a + θ/360) (coiling direction, N_a in turns, θ in degrees)
D_unwound = D × N_a / (N_a - θ/360)
Design implication: must check clearance to arbor or housing at maximum deflection
Body Length Under Load
L_loaded = (N_a × d) when coils are in contact
Active length may be constrained by arbor + end tang geometry
End Types
| Type | Stress | Notes |
|---|
| Straight torsion (tang) | High at bend | Simple; bending at angle |
| Hinged end (hook) | Moderate | Easier attachment |
| Short hook | Lower | Reduced stress raise |
| Straight offset | Variable | Depends on geometry |
Tang stress: σ_tang = 6Fa/d³ × 32/π where a = tang arm length
Allowable Stress
Static: σ_allow = 1.0 S_y (spring set occurs above yield; typical allowable = 0.78 S_u)
Per Shigley: S_y ≈ 0.78 S_u for cold drawn wire
Fatigue (inner fiber, N = 10⁷ cycles, music wire):
σ_allow,fatigue ≈ 0.50 S_u (estimated endurance limit for bending)
Use Goodman: σ_a/σ_e + σ_m/S_u ≤ 1
Wire S_u: same table as compression spring
Music wire (A228): S_u = 2061/d^0.163 [MPa, d in mm]
Design Procedure
- Set C (spring index), d, material
- Calculate k_θ, N_a from required rate and envelope
- Compute K_i, σ_max at maximum moment M_max
- Check σ_max vs. allowable
- Check coil diameter change vs. arbor clearance
- Check body length and tang geometry
- Calculate fatigue safety factor if dynamic
Fatigue Procedure
Mean moment M_m = (M_max + M_min)/2
Alternating M_a = (M_max - M_min)/2
σ_m = K_i × 32M_m/(πd³); σ_a = K_i × 32M_a/(πd³)
Goodman: σ_a/σ_e + σ_m/S_u ≤ 1
Material Reference
Same as compression springs; however E governs:
| Material | E [GPa] | G [GPa] |
|---|
| Music wire (A228) | 207 | 79 |
| Stainless 302 | 193 | 68 |
| Phosphor bronze | 103 | 44 |
Output
Provide: d [mm], D [mm], C, N_a, k_θ [N·mm/deg], σ_max [MPa] vs. σ_allow, coil diameter at max deflection [mm], arbor clearance, body length [mm], fatigue safety factor.
