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.
