Violin making · Material and geometry

Thickness, arching and modal frequency

Why does thinning change both mass and stiffness?

Geometry translates material properties into a structural response. A simple strip explains the strong role of thickness, while an arched orthotropic plate needs a richer model.

Try the interactive demo

Start with something familiar

A thin ruler bends more easily than a thick one of the same material. Removing thickness takes away mass, but removes bending resistance even faster. This is why carving cannot be understood as a mass adjustment alone.

Luthier Franz Übelhör uses a small finger plane to carve the inside of a violin back plate — Connect the diagram to a real object
Franz Übelhör works on the inside of a violin back with a finger plane. This is a real example of shaping a plate’s thickness distribution; the photograph does not specify a target thickness.
Kritzolina · Wikimedia Commons · CC BY-SA 4.0
Article photographs retain their full proportions; covers show a cropped view.

h = 100%

m = 100% · EI = 100.0%

h = 90%

m = 90% · EI = 72.9%

Diagram · Identical material and outline, different thickness. The percentages isolate the effect of geometry in an ideal uniform strip.

Notice what changes
For a uniform strip, 10% less thickness removes 10% of mass when length, width and density stay fixed. No new material property has been created.
Understand why
Bending rigidity scales with thickness cubed. Keeping 90% thickness leaves only 72.9% of that rigidity. Frequency combines both rigidity and inertia, so it follows a different scaling again.
Connect it to wood and making
A carved arched violin plate is not a uniform ruler. Thickness maps, arching, grain and assembly can redistribute motion; the strip teaches a mechanism rather than prescribing a final graduation.

Put names to the ideas

Mass and bending rigidity scale differently

For a uniform rectangular strip with fixed width and length, mass scales with h, while EI scales with h³. Removing a little thickness can therefore remove a larger proportion of bending rigidity.

Frequency is a combined result

The ideal strip has f ∝ h√(E′/ρ)/L². A frequency target alone cannot determine mass: different combinations of material and geometry can meet it.

Arching introduces another mechanism

An arched plate can couple bending and in-plane deformation. Thickness distribution, outlines, grain direction, bass bar, soundpost and assembly conditions change the resulting modes.

m\propto h,\quad EI\propto h^3,\quad f_{\mathrm{strip}}\propto\frac{h}{L^2}\sqrt{\frac{E'}{\rho}}

The symbol ∝ means “changes in proportion to” while other stated factors are fixed. Here mass follows h, bending rigidity h³, and ideal strip frequency h. These rules belong to the stated comparison.

A worked example

A 10% thickness reduction in the ideal uniform strip leaves 90% of mass, 72.9% of bending rigidity and 90% of its frequency. This scaling is not a carving prescription for a violin.

Explore it yourself

Compare material measurements before carving, then record the actual thickness map, arching and modal results after each controlled change.

Remaining thickness ratio h/h₀

Thickness retained: 90 %
Bending rigidity retained: 72.9 %

Uniform rectangular slender strip, fixed material, length, width and boundaries. This changes actual geometry, unlike the reading-error calculator; an arched plate needs a different model.

Check your understanding

After a 10% uniform thickness reduction, how much ideal bending rigidity remains?72.9%90%

Start with something familiar

Connect the diagram to a real object

h = 100%

h = 90%

Notice what changes

Understand why

Connect it to wood and making