Compliances of wood for violin top plates

Abstract

In the orthotropic approximation, nine independent elastic compliance elements are needed to describe the elastic behavior of wood. Many are difficult to determine experimentally, particularly in the audio frequency region, where their dynamic consequences are of interest to violin makers. This study computes the normal mode frequencies of a representative arched shell, using a commercial finite element package, in order to compute a ‘‘dependence matrix,’’ Daij =(sij/f a)(df a/dsij), expressing the relative fractional change of the ath normal mode frequency f a to the fractional change of the sij compliance element. Since the sum of Daij over all nine compliances is − 1/2 for each normal mode a, the relative importance of each compliance element can be read directly from Daij for a given set of compliances and a given plate geometry. This study concludes that only three compliance elements are of critical importance for violin top plates. One of the consequences is that the substitution of manmade composite materials with higher symmetry properties (and, hence, fewer compliance elements) for Norway spruce is feasible.