Lowest natural frequency of four aluminum rods against length, FEM points on the 1/L² beam formula line

Longer wind chime rods sound lower. So what happens to the pitch if you cut a rod to half its length?

The frequency does not double. It goes up four times, which is two octaves. I solved four aluminum rods of different lengths in NovaSolver Studio, a free finite element app, and compared them with the textbook beam formula.

The setup

  • Solid round aluminum rod, 10 mm in diameter
  • Lengths of 150, 200, 283 and 400 mm
  • No supports at all (a rod hanging on a thread is close to completely free)

The calculation gives the natural frequency: how many times per second the rod swings back and forth in each mode. That is a guide to the frequencies present in the sound.

Results

The reference is the Euler-Bernoulli formula for a free-free beam, with β₁L = 4.730:

f1​=2πL2(β1​L)2​ρAEI​​
Length Studio Formula Difference
150 mm 1,977 Hz 2,000 Hz −1.1%
200 mm 1,118 Hz 1,125 Hz −0.7%
283 mm 560 Hz 562 Hz −0.3%
400 mm 281 Hz 281 Hz −0.2%

The 400 mm rod is at 281 Hz and the 200 mm rod at 1,118 Hz: half the length, about four times the frequency.

To move by one octave (a factor of two in frequency), change the length by a factor of 1.41, the square root of two. Going from 200 mm to 283 mm took the rod from 1,118 Hz to 560 Hz, almost exactly half.

Why the square of the length?

Double the length of a rod and its mass doubles, while its bending flexibility goes up eight times. Heavier and softer both mean slower, so the frequency changes by 1/√(2×8), which is one quarter. Frequency is inversely proportional to length squared.

The shorter rods sit a little further from the formula because it assumes a rod that is long compared with its thickness. The 150 mm rod is only 15 diameters long, and Studio's value came out 1.1% below the formula.

Where to hang it

Mode shapes of the lowest two bending modes of the 283 mm rod, with theoretical node positions

In the lowest mode, two points on the rod do not move. For the 283 mm rod, Studio's mode shape puts them about 22.4% of the length from each end, close to the beam formula.

A thread attached there barely restrains that mode. On a 283 mm rod, that is about 63 mm from the end.

Why it sounds like a chime

A rod vibrates in higher modes at the same time as the lowest one. In Studio, the second bending mode came out at about 2.70 to 2.75 times the lowest frequency (the formula says 2.76).

On ideal strings and pipes open at both ends, the higher tones line up at nearly whole-number multiples: 2×, 3× and so on. A chime rod's modes do not, and that is part of why it sounds metallic.

These runs give only the frequencies and the mode shapes. They say nothing about how loud the rod is or how long it rings. Aluminum is taken as E = 69 GPa and density 2,700 kg/m³.

What I used

NovaSolver Studio is a free, open-source analysis app. I imported each rod's shape, entered the material, and solved each one on three mesh levels. For all four rods, the lowest natural frequency barely changed as the mesh got finer. Connected to an AI app over MCP, Studio can take these setup and re-solve requests in plain language. These runs used a local development build after v0.7.6 (2026-11-08).