Library / Mechanical / MEC-05
Mechanical
Natural frequency of a mass on a spring
Frequency at which a spring-mass system oscillates freely.
- Ref
- MEC-05
- Rev
- v1
- Field
- Mechanical
- Result
- fn [Hz]
fn in hertz, k stiffness in N/m, m mass in kg. The angular natural frequency is ω_n = √(k/m) in rad/s. Undamped system.
A mass on a spring bounces at its own natural rate. Stiffer springs make it bounce faster; heavier masses make it bounce slower.
Calculator
- Angular frequency ωn20 rad/s
- Period0.3142 s
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- Because of the square root, frequency changes slowly: to halve the frequency you need four times the mass or a quarter of the stiffness.
- Excitation near fn causes resonance and large vibration; isolation mounts need a natural frequency well below the disturbance.
- The static deflection under the weight, mg/k, gives the same frequency as 15.8/√(δ in mm) hertz, a useful field check.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| k Stiffness | 100 to 1,000,000 N/m | 0.5033 | 50.33 | 49.83 | +10% | +4.88 % |
| m Mass | 0.1 to 1,000 kg | 0.3183 | 31.83 | 31.51 | +10% | −4.65 % |
Sweep of Stiffness
| Stiffness (N/m) | Natural frequency (Hz) |
|---|---|
| 100 | 0.5033 |
| 316.2 | 0.895 |
| 1,000 | 1.592 |
| 3,162 | 2.83 |
| 10,000 | 5.033 |
| 31,620 | 8.95 |
| 100,000 | 15.92 |
| 316,200 | 28.3 |
| 1,000,000 | 50.33 |
Sweep one variable
Variables and how each one contributes
Natural frequency
What the equation gives you. Everything else feeds it.
Stiffness
Spring rate: force per unit deflection. Raises frequency with the square root: 4× the stiffness doubles the frequency.
Mass
Mass that vibrates. Lowers frequency with the square root: 4× the mass halves the frequency.
Worked example with the values above
- ωn = √(k/m) = √(4,000 / 10) = 20 rad/s.
- Divide by 2π to convert to hertz.
- fn = 3.183 Hz.
Assumptions and limits
- Linear spring, rigid mass, one degree of freedom.
- Damping neglected (small damping changes f only slightly).
- Spring mass much smaller than the supported mass.
Common mistakes
- Reporting ω in rad/s when hertz is asked for.
- Using mass in grams.
- Using the stiffness of one spring when two act in parallel.
Statistics and uncertainty
With a square root, the relative error is halved: δf/f ≈ ½ √[(δk/k)² + (δm/m)²]. A ±10% stiffness uncertainty, common for rubber mounts, gives about ±5% on frequency.
Where it is used
- Equipment mounting and shock isolation.
- Avoiding resonance in structures and shafts.
- Vehicle suspension design.
Maths you need
- Hooke’s law: spring force F = kx
- Frequency: cycles per second in hertz