Beam Natural Frequency⚠ unverified

Understanding the Parameters

• Bending stiffness $EI$ — the product of material stiffness $E$ and section stiffness $I$ is the beam's resistance to bending. It plays the role of the spring constant: higher $EI$ raises the frequency as $\sqrt{EI}$. Choosing a deeper section (larger $I$) is the usual way to push a resonance upward.

• Mass per unit length $m$ — the distributed inertia (kg/m), the dynamic counterpart of the mass in a mass–spring oscillator. Frequency falls as $1/\sqrt{m}$: heavier beams vibrate more slowly. Note it is mass per length, not total mass — a common unit slip.

• Span $L$ — dominates through $1/L^2$. This quadratic sensitivity means small changes in span (or adding an intermediate support to halve the effective span) shift the natural frequency strongly — the main design lever for avoiding resonance.

• The constant $\pi^2$ — the eigenvalue of the first mode for pinned–pinned ends. Other supports give other constants (fixed–fixed $\approx 22.4$, cantilever $\approx 3.52$ in the equivalent form), so the boundary conditions matter as much as the geometry.

Derivation (Approaching a Proof)

Free transverse vibration of a slender (Euler–Bernoulli) beam is governed by the partial differential equation obtained from combining the moment–curvature law with Newton's second law on a beam element:

$$EI\,\frac{\partial^4 v}{\partial x^4} + m\,\frac{\partial^2 v}{\partial t^2} = 0.$$

Seek separable harmonic solutions $v(x,t) = \phi(x)\,e^{i\omega t}$. Substituting gives an ordinary differential equation for the mode shape $\phi$:

$$\phi'''' - \beta^4 \phi = 0, \qquad \beta^4 = \frac{m\,\omega^2}{EI}.$$

For simply supported ends the deflection and moment vanish at both ends ($\phi = \phi'' = 0$ at $x=0,L$), which the sine modes $\phi_n = \sin(n\pi x/L)$ satisfy exactly, requiring

$$\beta L = n\pi, \quad n = 1, 2, 3, \dots$$

Substituting $\beta = n\pi/L$ into $\beta^4 = m\omega^2/EI$ and solving for $\omega$:

$$\omega_n = (n\pi)^2 \sqrt{\frac{EI}{m L^4}} = \frac{(n\pi)^2}{L^2}\sqrt{\frac{EI}{m}}.$$

The fundamental ($n=1$) is $\omega_1 = \dfrac{\pi^2}{L^2}\sqrt{\dfrac{EI}{m}}$. The frequencies scale as $n^2$ — the second mode is four times the first — a signature of bending (fourth-order) systems, unlike strings, whose overtones scale as $n$.

Dimensional check. $\left[\dfrac{1}{L^2}\sqrt{\dfrac{EI}{m}}\right] = \dfrac{1}{\text{m}^2}\sqrt{\dfrac{\text{Pa}\cdot\text{m}^4}{\text{kg/m}}} = \dfrac{1}{\text{m}^2}\sqrt{\dfrac{(\text{N/m}^2)\text{m}^4 \cdot \text{m}}{\text{kg}}} = \dfrac{1}{\text{m}^2}\sqrt{\text{m}^4/\text{s}^2} = \text{s}^{-1}$. ✓

History and Development

The Euler–Bernoulli beam-vibration equation dates to the 18th century (Euler, Daniel Bernoulli), with the full modal solution developed through the 19th century (Rayleigh, Theory of Sound, 1877). It underpins modal analysis in mechanical and civil engineering — from avoiding resonance in machine frames, rotor shafts, and turbine blades to designing footbridges against pedestrian-induced sway. Timoshenko later added shear and rotary-inertia corrections for stubbier beams and higher modes.

Related Concepts: Critical Speed Shaft, Natural Frequency Cantilever Beam, Rectangular Moment of Inertia, Moment Of Inertia I Beam, Beam Bending Stress

Notes: For the simply-supported fundamental mode with $m$ as mass per unit length. Divide by $2\pi$ for frequency in Hz. Euler–Bernoulli theory is accurate for slender beams; use Timoshenko theory for deep beams or high modes. Change the leading constant for other end conditions.