Spring Constants Reference
Hooke's law, helical coil spring formulas, series / parallel rules.
Reference
Core formulas
- Hooke's law
- F = −k · x
- Energy stored
- U = ½ k x²
- Natural frequency
- f = (1 / 2π) · √(k / m)
- Period
- T = 2π · √(m / k)
Helical compression spring
- Rate k
- = G · d⁴ / (8 · D³ · N)
- d
- Wire diameter
- D
- Mean coil diameter
- N
- Active coils
- G (modulus)
- ~80 GPa for music wire / spring steel
Combinations
- Series
- 1/k_total = 1/k₁ + 1/k₂ + … (softer than components)
- Parallel
- k_total = k₁ + k₂ + … (stiffer)
Material moduli (G, GPa)
| Material | G (GPa) |
|---|---|
| Music wire / spring steel | 80 |
| Stainless steel 302 | 70 |
| Beryllium copper | 47 |
| Phosphor bronze | 44 |
| Inconel X-750 | 75 |
| Titanium | 45 |
Notes
- Rate assumes elastic deformation — do not exceed yield stress or the spring takes permanent set.
- Fatigue matters for cyclic loads — stress ranges are specified in fatigue curves.
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