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Unique structural quality on the arching shape of belly and back

The Impact of Arching Shape on Structural Deflections (pdf - 2.4MB)


On this page, structural quality is shown that is determined by the earlier described arching structure. Based on geometry, the "Straight Tangent Lines" (STLs) are produced as seen in figure 1.

An STL is a line that intersects the iso lines with equal distance on the arching surface while at the same time the iso line levels have equal high distances. The figure shows STL state from 4 to 13mm level.

Figure 1, Iso Lines with STLs

Extending the shape of these STLs brings about the shape of a pyramid as seen in figure 2.

Figure 2, The framework

On the violin, the structures that arise resemble a pyramidal frustum, as seen in figure 3. The top of the pyramid is cut off producing an arching shape.

Extending the STLs on the exterior creates two pyramids. These have common base producing the shape of an octahedron, defined by all lines shown in grey.

Figure 3, The pyramidal frustum
of the belly and back

The adjacent animation shows the length cross section at the location of the sound post/bridge. When string load is applied the structure starts to deform. By clicking on the arrows in the box at the right we can observe what happens with the structure.

This can be done step by step (8 steps) or all steps at once (motion) in this deflecting process, neither the sound post nor the cross section in front of the upper F-hole, marked by triangles, move.

Looking at the deflection on each side of the sound post/bridge and the upper F-holes, we observe the upper and lower bout shapes to deflect equally on each side. On a real instrument, the sum of rigidness of the belly plus, the back and the rib structure are different. Therefore, we cannot expect them to have equal deflection and stress condition.

The animation shows the optimal state where the bout shapes have equal deflection. Technically the load condition on the belly is a continuous beam which extends over four supports, joined together by three spans. When the centre span is forced down we can observe the outer spans to move upward. In the case of the violin belly the supports on the end of the outer spans are forced upward and inward by string load.

Thus the consequence of string load forces the centre span to bend in a downward direction while the outer spans become forced upward.

Finally when string load is applied at pitch we find the instrument in its static state of equilibrium. There are different stress conditions on the belly by the stationary supports, on the sound post and the upper F-hole cross section, and on the stationary sound post on the back.

What we see are two deflection structures (instruments) that act differently with the applied string load and also will have different dynamical behavior. The dynamical behavior is a produced by interaction between the belly, rib and back structures on each side.

The report, "The impact of arching shape on structural deflection", describes the circumstances that arise based on a special geometric arching structure when the instrument comes into its static stress condition.


When we place a column between the tops of the pyramids a “Partial Stabilizing Framework” (PSF) arises. Lengthening the column increases the stress on the STLs, making them resistant to forces of deformation . This PSF is a significant quality that makes it possible to control the behavior of the surrounding arching. The arching becomes divided into sectors demarcated by the STLs. When the arching, in the sectors shapes become stressed by string load, the sector shapes will have their own stress quality.

The photos above show a model where the PSF is demonstrated showing STLs as cords. The sound post is placed on the centre line stretching the cords making a framework that divides the arching into four sector shapes.

© Copyright, Robert Zuger. All Rights Reserved.