Time-Domain Simulation using Component Mode Synthesis for a Piano

We present a physics based, time-domain numerical simulation framework that treats a full-scale grand piano as a comprehensively coupled vibroacous-tic system, encompassing nonlinear hammer–string interaction, geometrically nonlinear and lossy strings with anisotropic end supports, a three dimensional orthotropic wooden body with structural damping, and exterior acoustic radiation. The approach combines hybrid interface component mode synthesis (CMS) and Galerkin method for the strings with a finite element (FE) body model and a fast multipole boundary element method (FMBEM) for the surrounding air. The large coupled body–air subsystem is reduced via a complex mode CMS based on nonlinear eigenvalue analysis, yielding complex modal parameters at string support locations together with transfer functions to virtual microphones for efficient auralization. Manufacturing-induced residual stresses are explicitly incorporated through nonlinear static analysis of the assembly process, enabling system level causal links from design and fabrication parameters to radiated sound. To our knowledge, this work represents the first fully integrated time-domain piano simulation that simultaneously couples: (i) a nonlinear, double layer generalized Maxwell model of hammer felt; (ii) a spatial, bidirectionally coupled string model with stiffness, dispersion, and zig-zag/anisotropic boundary conditions at bridge and bearing; and (iii) a 3D body–air model reduced by complex modal synthesis, including residual stress effects. The method reconstructs driving point mobilities and perceptually salient phenomena—partial dispersion, polarization exchange, and “ringing”—and provides audible examples as supplementary material. Beyond piano acoustics, the pipeline constitutes a generalizable methodology for large scale, strongly coupled nonlinear struc-tural–acoustic simulations in engineering design, enabling parameter sensitive, time-domain auralization with tractable cost.