Spring Natural Frequency⚠ unverified

Understanding the Parameters

• Spring rate $k$ — the restoring stiffness (from Helical Spring Rate). A stiffer spring pulls the mass back harder, raising the frequency as $\sqrt{k}$. To raise a resonance, stiffen the spring or lighten the mass.

• Mass $m$ — the inertia being moved. A heavier mass is slower to reverse, lowering the frequency as $1/\sqrt{m}$. This is why heavy machinery on soft mounts has low resonant frequencies (good for isolation) and why lightening reciprocating parts raises the safe operating speed.

• The $2\pi$ — converts the angular natural frequency $\omega_n = \sqrt{k/m}$ (rad/s) into cyclic frequency $f$ (Hz, cycles/s). Vibration isolation is effective only above $\sqrt{2}\,f$, so a low $f$ widens the isolation band.

Derivation (Approaching a Proof)

Apply Newton's second law to the mass on the spring. Measuring displacement $x$ from the static equilibrium position, the only net dynamic force is the spring's restoring force $-kx$ (gravity is cancelled by the static deflection):

$$m\ddot{x} = -k x \;\Longrightarrow\; \ddot{x} + \frac{k}{m}x = 0.$$

This is the simple harmonic oscillator equation. Its solution is sinusoidal, $x(t) = A\cos(\omega_n t + \varphi)$, with angular frequency found by substitution:

$$\omega_n = \sqrt{\frac{k}{m}}.$$

Converting to cyclic frequency ($f = \omega_n/2\pi$):

$$f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}.$$

The frequency depends only on $k$ and $m$, not on the amplitude — the hallmark of a linear (isochronous) oscillator that makes it useful for timekeeping. Damping lowers the frequency slightly to $f_d = f\sqrt{1-\zeta^2}$, negligible for the light damping of metal springs.

Dimensional check. $\left[\sqrt{k/m}\right] = \sqrt{\dfrac{\text{N/m}}{\text{kg}}} = \sqrt{\dfrac{\text{kg/s}^2}{\text{kg}}} = \text{s}^{-1}$; dividing by $2\pi$ keeps s⁻¹ = Hz. ✓

History and Development

The simple harmonic oscillator emerged from the union of Hooke's law (1678) and Newton's second law (1687) and was developed into the theory of vibration by Euler, d'Alembert, and Rayleigh (Theory of Sound, 1877). The formula governs everything from pendulum-alternative spring timekeepers to modern seismic base isolation. For springs themselves, the coil surge frequency (a wave travelling along the spring) is a related resonance that limits high-speed valve-spring operation.

Related Concepts: Natural Frequency mass-spring, Helical Spring Rate, Spring Energy, Critical Speed Shaft, Helical Spring Deflection

Notes: Undamped natural frequency of a lumped spring–mass system; assumes a massless spring. If the spring's own mass is significant, add roughly one-third of it to $m$. Keep forcing frequencies away from $f$ (and its harmonics) to avoid resonance/surge.