VIBRATIONS OF SHALLOW SPHERICAL SHELLS AND GONGS:

Linear and nonlinear vibrations of shallow spherical shells with free edge are investigated experimentally and numerically and compared to previous studies on percussion instruments such as gongs and cymbals. The preliminary bases of a suitable analytical model are given. The prime objective of the work is to take advantage of the specific geometry of perfectly isotropic and homogeneous spherical shells in order to isolate the influence of curvature from other possible causes of nonlinearities. Hence, combination resonances due to quadratic nonlinearities are especially studied, for an harmonic forcing of the shell. Identification of excited modes is achieved through systematic comparisons between spatial numerical results obtained from a finite element modeling, and spectral informations derived from experiments. INTRODUCTION Vibrations of percussion instruments such as cymbals and gongs are essentially non-linear. This is due to the large deflections of the instrument, especially at the free edge, when struck by the blow of a mallet. Today, no theoretical model has been developed for these instruments and it is the aim of the present paper to contribute to filling this gap. Our investigations started with an experimental study on cymbals subjected to an harmonic excitation [1]. While keeping the excitation frequency fexc constant and increasing the amplitude, three distinct vibrational regimes were exhibited. The first one is periodic, defined by an harmonic spectrum governed by fexc. A first bifurcation leads to the apparition of other spectral components, of frequencies that are incommensurate with fexc, the resulting vibration being quasi-periodical. A second bifurcation leads to the third regime, which is chaotic. As a first attempt to write a physical model, a study was conducted on circular plates [2,3], which has led to a theoretical understanding of the periodic regime, validated by experiments. It was shown in particular that the problem can be modeled by a set of coupled second order nonlinear oscillators. The next step was then to study the quasi-periodical phase of the vibration. Our second study of percussion instruments was conducted on a large orchestral gong (or Chinese tam-tam) [4]. The same route to chaos than for the cymbals was observed, and the quasi-periodical regime was especially investigated. Two results suggested us to think that quadratic non-linearities are to be considered in the model. The first result is that the flexural motion of the gong shows a clear distorsion due to the presence of a quadratic nonlinearity in the oscillators. The second result is that the observed combination of resonances are governed by frequency relations of the form: m f n f exc f + ≈ or n 2f exc f ≈ (1) where is the excitation frequency, exc f m f and f are eigenfrequencies of the gong. These frequency relations where also observed in the case of the cymbal [1], and are typical of quadratic nonlinearities [5]. These quadratic nonlinearities are a consequence of the curved geometry of the gong [6]. As a consequence, the flat plate model of [2] is inadequate since it exhibits cubic nonlinearities only. Our present objective is to investigate the nonlinear properties of a perfectly curved structure, of geometry as close as possible to those of gongs and cymbals, in order to facilitate the derivation of a theoretical model. The difficulties with real gongs are due to both their particular geometry and structure inhomogeneities [4]. The selected structure is a spherical cap. A number of caps, with different thickness, diameter and curvature, were especially designed in order to examine the influence of geometrical parameters on the quadratic non-linear effects. The caps are made of brass and are assumed to be perfectly homogeneous. n The next section is first devoted to an experimental linear modal analysis of the spherical caps. It enables to make several comparisons, first with circular plates, and then with the gong studied in [4]. In the nonlinear regime, several combination of resonances are reported. They are found to be similar to those observed in gongs and cymbals. In section 3, a theoretical model for nonlinear transverse vibrations of a spherical cap with free edge is presented, in order to highlight the relevant parameters of the problem. The successive steps for the resolution of the problem are described. EXPERIMENTS The set-up is similar to the one used for the experimental study on gongs [4] and is extensively described in a more recent paper [3]. Only the main features are given here. The spherical shell is freely suspended by means of nylon threads (guitar strings). Is has been checked that the suspension does not alter significantly the values of eigenfrequencies and decay times. A small magnet is glued on the structure, and driven by a fixed coil.