Nonlinear generation of missing modes on a vibrating string

Abstract

The nonlinear transfer of energy among modes of different frequencies on a vibrating string is investigated both theoretically and experimentally. The nonlinearity is associated with the wellknown variation of string tension caused by the vibration modes, but it is essential that at least one of the end supports has finite mechanical admittance if there is to be any mode coupling. If the nonrigid bridge support has zero admittance in a direction parallel to the string, the coupling is of third order in the mode amplitudes. For a more realistic model in which the string changes direction as it passes over a bridge of finite admittance there are additional coupling terms of second order. The first mechanism gives driving terms of frequency 2On _____ COm where COn and COm are, respectively, the angular frequencies of the nth and mth modes present on the string, while the second mechanism gives driving terms of angular frequencies 2On and 2Om. Analysis shows that modes absent from the initial excitation of the string can be driven to appreciable amplitude by these mechanisms, reaching their maximum amplitude after a time typically of order 0.1 s. Modes that are in nearly harmonic frequency relationship behave simply but coupling of modes that are appreciably inharmonic may give rise to rapid amplitude fluctuations. A simple experiment with a wire deflected by a bridge of elastic cord and plucked so as to eliminate a particular mode from the initial excitation provided general semiquantitative confirmation of the theoretical predictions.

Plucked-string synthesis algorithms with tension modulation nonlinearity

“…In addition to the decay of fundamental frequency, the nonlinear vibration provides a mechanism for coupling of the vibration modes assuming that at least one of the end supports is not completely rigid [5]. This allows generation of harmonics that are initially missing due to initial conditions [5].…”

“…We also assume that the propagation speed of longitudinal vibration is much larger than that of transversal vibration so that the increase in tension caused by prolongation is immediately spread over the whole string. The tension along the string, F t , can be expressed as [5] F F ES…”

“…where y is the displacement of the string and x is the spatial coordinate along the string (see, e.g., [5]). The tension modulation is illustrated in Fig.…”

“…Perceptually, the two most important phenomena caused by tension modulation are the pitch variation depending on the vibration amplitude and the generation of missing harmonics due to nonlinear coupling of the vibration modes. While the nonlinearity has been studied both analytically and experimentally [1], [6], [5], to our knowledge the only synthesis model of a vibrating string incorporating tension modulation has been proposed by Karjalainen et al [2], who used a memoryless nonlinearity. Synthesis models have been presented previously for a nonlinearly terminated string [8] and for the trombone where nonlinear wave propagation takes place [7].…”

Non-linear mode coupling in symmetrically kinked bars

“…As a first step towards understanding this non-linear mode coupling and energy transfer, we began by demonstrating and analyzing the phenomenon for a flexible string passing over a non-rigid bridge [5]. In that system both frequency doubling and frequency tripling were observed and, since its mode frequencies are harmonic to a good approximation, energy transfer to missing modes on a timescale of about 0.1 seconds was observed and accounted for by the theory.…”