Natural Frequency Mass Spring⚠ unverified

Understanding the Parameters

• Stiffness $k$ — the restoring force per unit displacement. A stiffer spring pulls the mass back harder, so it oscillates faster: $\omega_n \propto \sqrt{k}$. Quadrupling stiffness only doubles the frequency.

• Mass $m$ — inertia resists acceleration, so a heavier mass oscillates slower: $\omega_n \propto 1/\sqrt{m}$. This is why adding mass (or a tuned absorber) lowers a structure's natural frequency.

• $\omega_n$ vs $f_n$ — this returns the angular frequency (rad/s); the ordinary frequency in Hz is $f_n = \omega_n/2\pi$ and the period is $T = 2\pi/\omega_n$.

• Undamped — damping shifts the free-vibration frequency slightly to $\omega_d = \omega_n\sqrt{1-\zeta^2}$ (Damped Natural Frequency), but for light damping ($\zeta < 0.1$) the difference is under 1 %, so $\omega_n$ is used throughout design.

Derivation (Approaching a Proof)

Apply Newton's second law to a mass $m$ on a spring $k$, measuring displacement $x$ from equilibrium. The spring exerts a restoring force $-kx$, so

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

This is the equation of simple harmonic motion, $\ddot{x} + \omega_n^2 x = 0$, whose solution is sinusoidal, $x(t) = X\cos(\omega_n t + \phi)$. Matching the coefficient identifies the angular frequency:

$$\omega_n^2 = \frac{k}{m} \;\Longrightarrow\; \omega_n = \sqrt{\frac{k}{m}}.$$

The frequency is a property of the system ($k$ and $m$) alone — independent of the amplitude or how it was started — which is the defining feature of a linear oscillator.

Dimensional check. $\omega_n = \sqrt{\dfrac{k}{m}} = \sqrt{\dfrac{\text{N}/\text{m}}{\text{kg}}} = \sqrt{\dfrac{\text{kg}/\text{s}^2}{\text{kg}}} = \sqrt{\dfrac{1}{\text{s}^2}} = \dfrac{1}{\text{s}}$ (rad/s), as required.

History and Development

The mass–spring oscillator is the archetype of vibration theory, tracing to Hooke's law (1678) and the harmonic-motion analysis of the 17th–18th centuries. Its natural frequency underlies everything from clock balance wheels to seismometers, engine mounts, and building dynamics. Real systems are reduced to equivalent mass–spring models (via Spring Natural Frequency, Beam Natural Frequency, or Critical Speed Shaft) so that this one formula, applied mode by mode, governs their resonances.

Related Concepts: Damped Natural Frequency, Damping Ratio, Magnification Factor, Spring Natural Frequency, Beam Natural Frequency, Critical Speed Shaft

Notes: Angular frequency (rad/s); $f_n = \omega_n/2\pi$ (Hz), $T = 2\pi/\omega_n$. $\omega_n \propto \sqrt{k/m}$ — stiffer = faster, heavier = slower. Undamped; damped free frequency $\omega_d = \omega_n\sqrt{1-\zeta^2}$ (negligible for $\zeta<0.1$). Sets resonance and the ratio $r=\omega/\omega_n$.