Wind Chime Length, Marimba Bar and Tubular Bell Calculator

The uniform model is the free-free Euler–Bernoulli beam:

The two zero-frequency rigid motions are excluded. The first elastic root is $\beta_1=4.730040745$, followed by 7.853204624 and 10.995607838. Solve this equation for frequency or for length. For a tube, $A=\pi(D^2-d^2)/4$ and $I=\pi(D^4-d^4)/64$; set $d=0$ for a rod. For a bar bending through thickness $h$, $A=bh$ and $I=bh^3/12$. Moor (2005), attached s3 p.5, gives the beam relation and rounded roots; p.6 supplies the generic material properties.

The exact uniform mode shape is $Y=\cosh(\beta s)+\cos(\beta s)-\sigma[\sinh(\beta s)+\sin(\beta s)]$, with $s=x/L$ and $\sigma=(\cosh\beta-\cos\beta)/(\sinh\beta-\sin\beta)$. Stable exponential forms avoid cancellation for high trial modes. The first root of $Y_1(s)$ is 0.224157522, consistent with the rounded 0.224 suspension rule.

Undercut model, derived here. The remaining thickness ratio is $r(s)=1-d_c\cos^2[\pi(s-1/2)/u]$ inside a centered scoop of length $uL$, and 1 elsewhere. Thus $A(s)=A_0r$ and $I(s)=I_0r^3$. Sixteen exact uniform bending modes are trial functions. Subtract their mass-weighted rigid translation and rotation components. Form $K_{ij}=\int_0^1r^3Y_i''Y_j''\,ds$ and $M_{ij}=\int_0^1rZ_iZ_j\,ds$ using composite Simpson quadrature with 640 intervals. A Cholesky reduction gives a symmetric generalized eigenproblem. Its square-root eigenvalues replace $\beta_n^2$; reconstruct the eigenvectors to locate the fundamental nodes. The retained mass fraction is $1-d_cu/2$.

Equal temperament uses $f=A_4 2^{(m-69)/12}$. Length sweeps hold the cross-section and fractional scoop geometry constant. The animation reads a separate cycle input after the modal solution has been calculated.

• Linear, small-displacement Euler–Bernoulli bending; free ends, with negligible support forces at the fundamental nodes.
• Uniform density and elastic modulus. Circular tubes have concentric bores. Rectangular bars have constant width, at least as large as the full thickness, and bend through their stated thickness. Higher modes in the table belong to this bending plane; orthogonal bending modes are not included.
• The scoop is symmetric, on the underside, and follows the stated cosine-squared profile. Its depth and span are fractions; changing overall length rescales the scoop too.
• Material presets are generic values from Moor, not alloy or timber specifications. Custom properties are permitted; wood requires an effective longitudinal modulus, but orthotropic coupling and humidity changes are omitted.
• The numerical undercut formulation and drawings are derived here; they do not reproduce Lee Hite’s spreadsheet implementation or another author’s figure.

Use as a starting cut and tuning estimate, then verify the actual piece. The default tube has length/diameter about 20.3. A warning appears below 15; this screening ratio is an engineering choice, not a guaranteed accuracy threshold. Bars shorter than four widths may need a plate or solid model.

Undercutting is limited to 0–50% depth and a scoop spanning 40–80% of the length. An independent cubic-Hermite finite-element calculation with 80 and 160 elements was used as a numerical cross-check: the 16-mode result differed by less than 0.05% in the first three frequencies at the tested deepest/narrowest case (depth 0.5, span 0.4), and by less than 0.002% at depth 0.35, span 0.5. This checks the ideal beam solution, not manufacturing or real-instrument accuracy. A steep-scoop warning is triggered when maximum profile slope exceeds 0.15.

Lee Hite’s tables are rounded and use material constants that can differ from generic presets. Penn State’s published COMSOL example does not state its elastic properties; its reference case therefore uses a documented 2% comparison allowance. Do not infer a precision guarantee from a passing check.

Shear deformation, rotary inertia, damping, end caps, large holes, attached masses, tapered tubes, torsion, shell and plate modes, resonator air columns and temperature-dependent properties are not included. In a real tubular bell, perceived strike pitch can differ from the lowest bending mode. The animation is a normalized mode shape at a deliberately slow speed.

Removing material at the center generally lowers the fundamental; shortening uniform stock raises it. The plotted $f\propto L^{-2}$ law is not a trimming law for an already scooped bar whose absolute scoop is held fixed. Yamaha’s instrument guide (attached s4) describes fundamental, fourth and tenth harmonic tuning for marimbas. A single scoop need not achieve this 1:4:10 target. Measure the complete spectrum and refine the profile if harmonic tuning is required.