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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