Viscoelastic internal damping finite-difference model for musical instruments physical model sound production

Abstract

A viscoelastic model for the internal damping of musical instrument parts, like membranes or plates is implemented within a Finite-Difference Time Domain (FDTD) method. Internal damping of wood, leather, nylon, mylar, glue, or varnish strongly change the timbre of musical instruments and the precise spectrum of this damping contributes strongly to the individual instrument character. The model assumes a complex, frequency-dependent and linear stiffness in the frequency domain, which is analytically transferred into the time-domain using a Laplace transform. The resulting mass-weighted restoring force integral of the respective membrane or plate differential equation is solved using a circular buffer accumulation method for each spatial node on the geometry, which is effective, as the model is implemented on a massive parallel Graphics Processing Unit (GPU). The model is able to reproduce arbitrarily shaped internal damping frequency responses with sharp bandwidth and fast response. The model is also able to reproduce other energy distribution problems, like energy loss or even energy supply by different parts of musical instruments through coupling, time-dependent energy loss and supply behavior, or nonlinear damping behavior, like amplitude-dependent loss strength. So also internal damping of metamaterials can be calculated with this model.