Bridging the Gap Between Theory and Computation: Sparse Polynomial Chaos Modeling of Uncertain Wood Properties in Musical Instrument Soundboards

The rectangular plate with holes (RPwH) is a key component of many wooden string musical instruments, such as the guitar, fiddle, tamboura, and oud. One of the most important challenges in modeling wooden stringed musical instruments is the uncertainty arising from the mechanical properties of the materials. The sound produced by stringed instruments is designed by the vibrations of thin wooden shells and the soundboard (SB) is a critical component. In this study, the orthotropic mechanical properties of SBs are addressed, and their distributions are detected based on data sets and modeled to better account for uncertainty in statistical settings. The values of the flexural rigidities appeared in the well-known partial differential equation (PDE) for the calculation of the natural frequencies are assumed to be random variables with a distribution. The SB model is first approximated to a RPwH and then to a regular rectangular plate (RP). Deviations between the analytical solution and simulation results are based on the random variables. A polynomial chaos expansion (PCE) is constructed for the solution. Although there is no closed-form solution for the SB with holes, the so-called sparse PCE with a copula structure between these three geometries is obtained with a very small error value. The first 100 natural frequencies were computed for the RP, the RPwH, and the SB, with deviations observed between simulation and PDE solution results. The mean value of the PCE model was found to be 700 with a standard deviation of 1,349 and a coefficient of variation of 171.5%. The proposed analytical modeling framework can be used for uncertainty quantification and validation in any study of orthotropic RPs.