VIBRATIONS OF SHALLOW SPHERICAL SHELLS AND GONGS: A COMPARATIVE STUDY

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1 VIBRATIONS OF SHALLOW SPHERICAL SHELLS AND GONGS: A COMPARATIVE STUDY PACS REFERENCE: Kk Antoine CHAIGNE ; Mathieu FONTAINE ; Olivier THOMAS ; Michel FERRE ; Cyril TOUZE UER de Mécanique, ENSTA Chemin de la Hunière 976 PALAISEAU Cedex FRANCE Tel: (+33) (0) Fax: (+33) (0) chaigne@ensta.fr ABSTRACT Linear and nonlinear vibrations of shallow sherical shells with free edge are investigated exerimentally and numerically and comared to revious studies on ercussion instruments such as gongs and cymbals. The reliminary bases of a suitable analytical model are given. The rime objective of the work is to take advantage of the secific geometry of erfectly isotroic and homogeneous sherical shells in order to isolate the influence of curvature from other ossible causes of nonlinearities. Hence, combination resonances due to quadratic nonlinearities are esecially studied, for an harmonic forcing of the shell. Identification of excited modes is achieved through systematic comarisons between satial numerical results obtained from a finite element modeling, and sectral informations derived from exeriments. INTRODUCTION Vibrations of ercussion instruments such as cymbals and gongs are essentially non-linear. This is due to the large deflections of the instrument, esecially at the free edge, when struck by the blow of a mallet. Today, no theoretical model has been develoed for these instruments and it is the aim of the resent aer to contribute to filling this ga. Our investigations started with an exerimental study on cymbals subjected to an harmonic excitation []. While keeing the excitation frequency f exc constant and increasing the amlitude, three distinct vibrational regimes were exhibited. The first one is eriodic, defined by an harmonic sectrum governed by f exc. A first bifurcation leads to the aarition of other sectral comonents, of frequencies that are incommensurate with f exc, the resulting vibration being quasi-eriodical. A second bifurcation leads to the third regime, which is chaotic. As a first attemt to write a hysical model, a study was conducted on circular lates [,3], which has led to a theoretical understanding of the eriodic regime, validated by exeriments. It was shown in articular that the roblem can be modeled by a set of couled second order nonlinear oscillators. The next ste was then to study the quasi-eriodical hase of the vibration. Our second study of ercussion 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-eriodical regime was esecially 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 resence of a quadratic nonlinearity in

2 the oscillators. The second result is that the observed combination of resonances are governed by frequency relations of the form: f exc f n + f m or f () exc f n where f exc is the excitation frequency, f m and f n are eigenfrequencies of the gong. These frequency relations where also observed in the case of the cymbal [], and are tyical of quadratic nonlinearities [5]. These quadratic nonlinearities are a consequence of the curved geometry of the gong [6]. As a consequence, the flat late model of [] is inadequate since it exhibits cubic nonlinearities only. Our resent objective is to investigate the nonlinear roerties of a erfectly curved structure, of geometry as close as ossible 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 articular geometry and structure inhomogeneities [4]. The selected structure is a sherical ca. A number of cas, with different thicess, diameter and curvature, were esecially designed in order to examine the influence of geometrical arameters on the quadratic non-linear effects. The cas are made of brass and are assumed to be erfectly homogeneous. The next section is first devoted to an exerimental linear modal analysis of the sherical cas. It enables to make several comarisons, first with circular lates, and then with the gong studied in [4]. In the nonlinear regime, several combination of resonances are reorted. 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 sherical ca with free edge is resented, in order to highlight the relevant arameters of the roblem. The successive stes for the resolution of the roblem are described. EXPERIMENTS The set-u is similar to the one used for the exerimental study on gongs [4] and is extensively described in a more recent aer [3]. Only the main features are given here. The sherical shell is freely susended by means of nylon threads (guitar strings). Is has been checked that the susension 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. Modal analysis Figure : Geometry of the shell. Left: cross-sectional view. Right: to view. Shell index Radius of curvature Thicess h X= a /Rh R (mm) (mm) Table : Geometrical and elastic arameters of the shells. X is a nondimensional coefficient. The geometries of the four shells used in our exeriments are shown in Fig. and Tab.. The eigenfrequencies of the shells are measured by exciting the structure with a filtered white noise. They are comared to those derived from a theoretical model of sherical shells with free-edge found in [7]. These eigenfrequencies are written:

3 ù ( í ) a ~ = ù + X with Rh ì 4 X = () In Equation (), a is the radius, R the radius of curvature and h the thicess are (Fig. ). denotes the eigenfrequency of the shell mode with k nodal diameters and n nodal circles, denotes the corresonding eigenfrequency of the circular late of same radius a. The coefficients µ are the nondimensional numbers corresonding to the eigenvalues for a circular late with free edge [8,]. One can see in Equation () that the eigenfrequencies of the shells differ from the ones of the corresonding late through the nondimensional geometrical arameter X only. In articular, Equation () gives the eigenfrequencies of a thin circular late when X=0 ( R + ). In the analytical rocedure, the frequencies have been first calculated nondimensionally. The density is measured in a straightforward way. The Young s modulus corresonds to a standard value for brass. The value of the thicess h is adjusted so that the theoretical frequencies fit the exerimental ones. The values are listed in Table. Table 3 gives the values of the coefficients µ and the theoretical and measured shell eigenfrequencies ω, for shell number referenced in Table. Material Young s mod. Density r Radius a Poisson s E (N/m ) (kg/m 3 ) (mm) coefficient n Brass Table : Measured arameters of the shell. Mode index Circular late coefficient m Theoretical freq. Measured freq. (,0) ? (3,0).0.4? (4,0) and 35.6 (5,0) and 58.4 (6,0) and 83.9 (7,0) and.9 (8,0) and 4.0 (9,0) and 3.4 (0,0) and 50. (,0) and 95. (,0) and (0,) (0,) Table 3: Theoretical and measured eigenfrequencies of sherical shell Nr (X=a /Rh=66). The first index refers to the number of nodal radii, while the second indicates the number of nodal circles. In contrast to the case of gongs, Table 3 shows that the frequencies of a relatively large number of asymmetric modes are smaller than the lowest axisymmetric mode (0,). This roerty is a consequence of essentially two facts: a) The edge of the shell is free, and thus its transverse motion is significantly less constrained than in the case of a gong with a stiff ring [4]. b) The geometry of the shell is curved, which yields more stiffness than for the gong, esecially in its central art. This roerty is also resonsible for the fact that asymmetric modes with at least one nodal circle ( k 0, n ) have eigenfrequencies lower than f 0,. The asymmetric modes are groued here by airs, for which both configurations have almost identical eigenfrequencies. The corresonding modal shaes have the same attern, with a quadrature satial hase shift (see Fig. ). This secial feature comes from the fact that in a ù ù ~

4 erfect structure of revolution, each asymmetric eigenfrequency degenerates into two indeendent modes [9,]. The slight differences between the measured frequencies of both configurations in the case of our sherical cas (see Table 3), are due to small imerfections, mainly induced by the nylon threads. They aear to be negligible with resect, for examle, to the differences measured in the case of our gong [4]. This can be exlained by the fact that the gong shows structural and material inhomogeneities due to hammering and thermic treatment during its making [4]. Taking into account the existence of two configurations for the asymmetric modes is essential, as these configurations are indeendently taking art in the combination of resonances which are usually observed [4,]. Table 3 shows an almost erfect agreement between calculated and measured eigenfrequencies excet for the lowest axisymmetric mode. This effect might be due to the resence of the magnet (0.5 g) glued at the center of the shell. However, more systematic exeriments and calculations are needed here in order to validate this assumtion. Figure. From left to right: the two configurations of the (6,0) asymmetric mode, and modal shae of the axisymmetric (0,) mode for the sherical ca. Combination of resonances in the non-linear regime. Forcing frequency combination of resonances subharmonics (articular case) 4 (0,) (7,0) 354 (0,) 38 (,0) shifted+ 35 (4,0) 47 (0,0)+ 07 (7,0) shifted (9,0)+ 43 (8,0) 444 (0,3) (0,) 336 (,0) + 08 (7,0) shifted Table 4: Summary of the most rominent exerimentally observed combination of resonances. Table 4 shows some examles of observed combination of resonances, for the ca Nr (see Table ) excited at its center with an harmonic force of frequency successively adjusted to one articular axisymmetric eigenfrequency. All combinations of resonances in Table 4 corresond to the two cases resented in Equation (). Fig 3 shows the velocity sectrum of the ca excited at 354 Hz (f 0 ). These henomena are similar to those observed in cymbals [] and gongs [4]. Figure 3: Velocity sectrum of one selected oint on the ca. Fexc=348 Hz. The resonances excited through quadratic nonlinearities (see Table 4) are clearly visible.

5 THEORY Governing equations The nonlinear artial differential equations in olar coordinates for the sherical ca are the following [0]: DÄ w L(w,F) + ÄF + ñh w& = (r,è, t) R (3) Eh Eh Ä F + L(w, w) Äw = 0 R (4) F,r F,èè w,r w, w,rè w F,rè F,è L(w, F) = w + + +,rr F,rr r r r r r r r r (5) 3 Eh D = ( í ) (6) In Equations (3)-(6), w refers to the vertical transverse dislacement, F is the Airy stress function and (r, è, t) is the external ressure. D is the flexural rigidity. The underlying assumtion are that w is of the order of the shell thicess, and that the shell is shallow (h<<a<<r). Letting R + in Eqs (3)-(4) yields the nonlinear equations for a nonlinear late []. The case L = 0 corresonds to the linear sherical shallow shell model. Both conditions together lead to the well-own linear late equations. The curvature factors (roortional to /R in Eqs. (3) and (4)) are resonsible for a linear couling between the transverse dislacement w and the longitudinal stretching of the shell elements (which is a function of F). Equation (5) shows that L (w, w) is a quadratic function of the dislacement w. As a consequence, the Airy F/ R function F is both linear and quadratic in w through (3). Similarly, is a quadratic function of the dislacement in (), whereas L (w, F) has quadratic and cubic terms in w. This imortant feature illustrates the fact that both cubic and quadratic terms necessary coexist in the nonlinear sherical shell equations, in contrast to the nonlinear late equations which contain cubic terms only. To summarize this discussion, one can say that the quadratic nonlinearity is due to the coexistence between the nonlinear function L and the linear curvature terms. Modal rojection and resolution. The solution w is exanded onto the linear eigenmodes of the shell: + = w(r, è, t) = Ö (r,è).q (t) (7) where Ö is the th modal shae. With little mathematics [], one can show that the time functions q are solutions of a infinite set of nonlinear second-order differential equations, couled by quadratic and cubic terms: ä q& + ù q = á uvquq v u= v= s= u= v= q & à q q q + Q (8) suv s u v In Equation (8), ä and resectively. Coefficients Q denote the daming coefficient and the forcing of the th mode, á uv and à suv are functions of the modal shaes The next ste consists of truncating the infinite set of Equation (8) by assuming that the forced vibration of the shell is governed by a finite number of modes only [5]. The quasi-eriodic regime corresonding to a subharmonic resonance between mode (0,) and mode (7,0), for examle, is governed by three oscillators: the first oscillator corresonds to mode (0,) (of frequency f 0 = 4Hz Ö. ), and the two other oscillators corresond to the two configurations of

6 mode (7,0) (of frequencies f 70 = 0.8Hz and f =.9 70 Hz, resectively), The vibration shown in Figure 3 can be described by at least a set of five oscillators, sine at least one axisymmetric and two asymmetric modes are nonlinearly couled. The resolution of the resulting roblems for a finite number of oscillators can be either erformed by aroximate analytical methods such as the multile scale method [5], or by a direct numerical resolution (using a Runge-Kutta algorithm, for examle). This work is currently in rogress. CONCLUSION So far, our study on the large magnitude forced oscillations of thin shallow sherical shells with free edge can be summarized as follows: Like for the gong, combination resonances dues to quadratic nonlinearity are resent. These resonances are governed by frequency laws, such as Eq. (). For thicess, radius and diameter comarable to gongs, the rank of the lowest axisymmetrical modes is significantly higher for a sherical shell. This roerty is due to the free edge of the shell, comared to the relatively stiff edge of gongs, and to the more ronounced curvature of the ca. In ractice, this means that a significant number of combination resonances can be found here for the (0,) and (0,) mode, whereas one had to excite at frequencies at least equal to the (0,3) mode of the gong in order to obtain a large number of combinations. For sherical shells with free edge, the lowest frequencies are essentially asymmetric (n,0) modes. The homogeneity of the shell facilitates the analysis in the sense that each asymmetric mode is characterized by a doublet of frequencies with close values. This analysis is confirmed by numerical exeriments. In addition to the interest of comaring sherical shells with gongs, in the context of musical acoustics, our measurements contribute to yield exerimental and numerical data on shells with free edge, for which there are very few ublished aers, as far as we are aware. The nonlinear henomena encountered in gongs and cymbals, and more generally in thin structures, are due to geometrical nonlinearities, and directly linked to the large amlitude of the vibrations (of the order of the thicess, or even larger. In the articular case of sherical shells, it has been shown in Equation (8) that the curvature is resonsible for the resence of the quadratic terms. From the resent comarison between nonlinear behavior of shells and gongs, which shows high similarity in the nonlinear quasieriodical regime, one can reasonably claim that the imerfections observed in gongs have relatively no effect on the nonlinear behavior of these instruments comared to the curvature. REFERENCES [] Touzé, C., Chaigne, A. Lyaunov exonents from exerimental time series : alication to cymbal vibrations, Acta Acustica (000), 86, [] Touzé C., Thomas O., and Chaigne, A. Asymmetric non-linear forced vibrations of free-edge circular lates. I. Theory, J. Sound Vib. (00), to be ublished. [3] Thomas O., Touzé, C., and Chaigne, A. Asymmetric non-linear forced vibrations of freeedge circular lates. II. Exeriments, J. Sound Vib. (00), submitted. [4] Chaigne, A., Touzé, C., and Thomas, O. Nonlinear axisymmetric vibrations of gongs, Proceedings of ISMA 00, Perugia, 47-5 (00). [5] Nayfeh, A. H., Mook, D. T., Nonlinear oscillations, Wiley-Interscience, New York (979). [6] Thomas, O., Touzé, C. and Chaigne, A. Nonlinear behavior of gongs through the dynamics of simle rods systems, Proceedings of ISMA 00, Perugia, (00). [7] Johnson, M. W. and Reissner, E. On transverse vibrations of shallow sherical shells, Quart. Al. Math. (956), 5(4), [8] Leissa, A. W., Vibrations of lates, Acoustical Society of America, 993. [9] Meirovitch, L. Analitycal methods in vibrations. MacMillian Publishing Co. (967) [0] Leissa, A. W., and Kadi, A. S., Curvature effects on shallow shell vibrations, J. Sound Vib.(97), 6(), [] Yasuda, K. and Kushida, G. Nonlinear forced oscillations of a shallow sherical shell. Bulletin of the JSME (984), 7(3),