Nonlinear vibrations and chaos in gongs and cymbals

Abstract

This paper summarizes some results obtained in the last few years for the modeling of nonlinear vibrating instruments such as gongs and cymbals. Linear, weakly nonlinear and chaotic regimes are successively examined. A theoretical mechanical model is presented, based on the nonlinear von Kármán equations for thin shallow spherical shells. Modal projection and Nonlinear Normal Mode (NNM) formulation leads to a subset of coupled nonlinear oscillators. Current developments are aimed at using this subset for sound synthesis purpose.

Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates

“…2 and its dimensional counterpartF readsF = Eh 4 εa 4 F. For a typical circular plate having the dimension of a cymbal, i.e. with E=110 GPa, h=1 mm, a= 0.2 m and ε = 12(1 − ν 2 ) with ν = 0.33, the critical non-dimensional force F cr = 15.92 leads to a dimensional forcingF=102.35 N, which is significantly larger than the experimental values needed to obtain chaotic behaviour, which are of the order of 2 to 10 N [8,50]. In section 4, it will be shown that considering an imperfection (unavoidable in real plates) significantly reduces this critical value to those observed experimentally.…”

“…A convenient and reproducible way to experimentally study the chaotic vibrations of gongs and cymbals consists in harmonically forcing the structure with an increasing amplitude. Numerous experiments on differents cymbals and gongs have been performed and are reported in [9,7,10,8,11,12]. The generic observation, that is also valid for plates and shells, reveals a scenario for the transition to chaotic vibrations including two bifurcations, separating three distinctive regime.…”

“…The study of the transition from periodic to chaotic vibrations is of primary interest in numerous applied fields such as aeronautic and aerospace or civil engineering, where shell-like structural components are often used [1,2]. Another field where chaotic vibration of shells is searched for, is that of musical acoustics and more precisely the sound of gongs and cymbals where the chaotic nature of the vibration ensures for the peculiar bright and shimmering sound of these instruments [3,4,5,6,7,8]. A convenient and reproducible way to experimentally study the chaotic vibrations of gongs and cymbals consists in harmonically forcing the structure with an increasing amplitude.…”

“…This quasiperiodic state is characterized by the appearance of a number of distinct frequency peaks in the spectrum, each of them being an eigenmode of the structure. Moreover, these new frequency peaks can be grouped by two or three, and they present simple nonlinear resonance relationships with the excitation frequency [9,7,10,8]. Hence, this transition is interpreted as a loss of stability of the first excited mode, in favour of a coupled regime, involving all the modes that share an internal resonance relationship with the excitation frequency, the injected energy being spread all over this specific set of modes.…”

“…Hence, this transition is interpreted as a loss of stability of the first excited mode, in favour of a coupled regime, involving all the modes that share an internal resonance relationship with the excitation frequency, the injected energy being spread all over this specific set of modes. Finally, for high values of the forcing amplitude, a second bifurcation occurs and a chaotic vibration, characterized by a broadband Fourier spectrum 2 and a positive Lyapunov exponent, sets in [7,8].…”