Mass-Spring Resonators

Since a harmonic oscillator is produced by a simple mass-spring system, a mechanical generator for the harmonic basis functions of Bernoulli is readily obtained by equating Newton's second law $f=m\ddot{x}$ for the reaction force of an ideal mass $m$, with Robert Hooke's spring force law $f=-kx$ (published in 1676), where $k$ is an empirical spring constant [62]. Hooke (1635-1703) was a contemporary of Newton's who carried out extensive experiments with springs in search of a spring-regulated clock [243, pp. 274-288]. Hooke's law was generalized to 3D by Cauchy (1789-1857) as the familiar linear relationship between six components of stress and strain.^D.7

Elementary mass-spring models have found much use in computational physical models for purposes of sound synthesis [66,87]. For example, a mass-spring oscillator is typically used to model a brass-player's lips [6], piano hammers [41], and is sometimes included in woodwind-reed models [379].