Real-time simulation of the struck piano string with geometrically exact nonlinearity via novel quadratic Hamiltonian method

Nonlinearities are key to many acoustic phenomena. Piano string vibration is a master example: nonlinearities are presentboth in the string itself (in the form of amplitude-dependent relationships) as well as in the hammer-string interaction. All such nonlinearities yield a number of perceptually meaningful phenomena, and must be accounted for at thesimulation stage. Though simulation of piano string vibration has been approached by means of various models,the numerical schemes for the geometrically exact nonlinearity are somewhat inefficient in the current landscape, relyingon expensive iterative procedures. Here, a novel discretisation method, based on a quadratic Hamiltonian, is given.The method is energy-preserving (and thus stable) by construction, yet it yields and update equation solvable withoutrecourse to iterative routines. Furthermore, the update matrix is invertible by means of a fast analytical formula, yieldinga sparse, explicit system to solve at each time step. With respect to previous schemes, the proposed schemes areorders-of-magnitude faster: for the first time, simulations of low-pitched piano strings, including both geometrically exactand contact nonlinearities, are attainable under real-time on standard consumer hardware.