Analyse et modélisation de vibrations non-linéaires de milieux minces élastiques - Application aux instruments de percussion

Les vibrations des instruments de percussion non-lineaires (famille regroupant les gongs et les cymbales) presentent des comportements complexes et des caracteristiques propres aux systemes non-lineaires en conditions normales de jeu. On peut notamment citer le spectre continu et a large bande en vibrations de grande amplitude et la sensibilite aux conditions initiales. Le but de cette etude est de fournir un debut de modelisation de ces structures, en vue de comprendre les mecanismes de generation du son et de contribuer a l'etablissement d'un modele pour la synthese sonore. Dans une premiere partie sont presentes des resultats experimentaux conduits sur un tam-tam chinois, lorsque celui-ci est excite a l'une de ses resonances par une force harmonique. Cela permet d'identifier des phenomenes non-lineaires simples, comme une dependance en l'amplitude des vibrations de la frequence des oscillations libres, des resonances hysteretiques associees a des phenomenes de saut entre differents regimes de vibration, et des echanges d'energie entre modes propres resultant de combinaisons de resonances. Une deuxieme partie est consacree a la modelisation des instruments de percussion par des plaques circulaires elastiques a bord libre, en grand deplacement et rotations moderees. Le modele non-lineaire de Von-Karman est utilise et son domaine de validite est discute. Les equations non-lineaires aux derivees partielles sont ensuite projetees sur la base des modes propres. Le cas particulier ou la vibration n'est gouvernee que par un mode est etudie en detail: un modele analytique de comportement est etabli et valide par une etude experimentale. Lorsque le mode considere est asymetrique, le couplage entre les deux modes propres degeneres est predit quantitativement de facon tres precise, au moyen d'un recalage du modele. Dans une troisieme partie, on etudie des systemes a un degre-de-liberte, composes de barres elastiques articulees. Une interpretation physique des non-linearites geometriques est tout d'abord proposee. Ces modeles permettent ensuite d'expliquer les comportements unimodaux du tam-tam. L'influence de la geometrie de la structure (a travers sa courbure et son epaisseur) sur l'incurvation des courbes de resonance et la distortion harmonique des signaux de vibrations est clairement etablie.

[1]  R. White,et al.  The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures part I: Simply supported and clamped-clamped beams , 1991 .

[2]  S. Timoshenko,et al.  LXVI. On the correction for shear of the differential equation for transverse vibrations of prismatic bars , 1921 .

[3]  T. D. Burton,et al.  Analysis of non-linear autonomous conservative oscillators by a time transformation method , 1983 .

[4]  M. R. Silva,et al.  Nonlinear Flexural-Flexural-Torsional Dynamics of Inextensional Beams. I. Equations of Motion , 1978 .

[5]  Staffan Schedin,et al.  Transient wave response of a cymbal using double-pulsed TV holography , 1998 .

[6]  R. Benamar,et al.  The Effects of Large Vibration Amplitudes on the Mode Shapes and Natural Frequencies of Thin Elastic Structures, Part II: Fully Clamped Rectangular Isotropic Plates , 1993 .

[7]  Gerhard A. Holzapfel,et al.  Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science , 2000 .

[8]  Ali H. Nayfeh,et al.  A numerical-perturbation method for the nonlinear analysis of structural vibrations , 1974 .

[9]  J. E. Gordon,et al.  Structures: Or Why Things Don't Fall Down , 1978 .

[10]  Ali H. Nayfeh,et al.  Modal interaction in circular plates , 1990 .

[11]  Pedro Ribeiro,et al.  The second harmonic and the validity of Duffing’s equation for vibration of beams with large displacements , 2001 .

[12]  Arthur W. Leissa,et al.  Vibration of Plates , 2021, Solid Acoustic Waves and Vibration.

[13]  Charles M. Krousgrill,et al.  Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance , 1993 .

[14]  R. D. Mindlin,et al.  Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates , 1951 .

[15]  G. C. Kung,et al.  Nonlinear Flexural Vibrations of a Clamped Circular Plate , 1972 .

[16]  A. H. Nayfeh,et al.  Non-linear resonances in the forced responses of plates, part 1: Symmetric responses of circular plates , 1975 .

[17]  Noboru Yamaki,et al.  Influence of Large Amplitudes on Flexural Vibrations of Elastic Plates , 1961 .

[18]  C. Chia.  Nonlinear analysis of plates , 1980 .

[19]  L. Meirovitch.  Analytical Methods in Vibrations , 1967 .

[20]  G. J. Efstathiades.  A new approach to the large-deflection vibrations of imperfect circular disks using Galerkin's procedure , 1971 .

[21]  A. Chaigne,et al.  Asymmetric non-linear forced vibrations of free-edge circular plates. Part II: Experiments , 2003 .

[22]  Ali H. Nayfeh,et al.  Nonlinear analysis of the forced response of structural elements , 1974 .

[23]  S. A. Tobias,et al.  Forced Undamped Non-Linear Vibrations of Imperfect Circular Discs , 1963 .

[24]  Arthur W. Leissa,et al.  Curvature effects on shallow shell vibrations , 1971 .

[25]  T. Kármán.  Festigkeitsprobleme im Maschinenbau , 1907 .

[26]  Christophe Pierre,et al.  Normal modes of vibration for non-linear continuous systems , 1994 .

[27]  Lawrence W. Rehfield,et al.  Nonlinear free vibrations of elastic structures , 1973 .

[28]  T. D. Burton,et al.  On the multi-scale analysis of strongly non-linear forced oscillators , 1986 .

[29]  J. Irons,et al.  Non-linear vibration of centrally clamped thin discs , 1989 .

[30]  Yi‐Yuan Yu,et al.  Nonlinear Vibrations of Shallow Spherical Shells , 1969 .

[31]  Neville H Fletcher,et al.  Nonlinearity, chaos, and the sound of shallow gongs , 1989 .

[32]  Functional analysis and the method of harmonic balance , 1975 .

[33]  T. Rossing,et al.  Analysis of cymbal vibrations using nonlinear signal processing methods , 1998 .

[34]  Joseph P. Cusumano,et al.  Chaotic non-planar vibrations of the thin elastica: Part I: Experimental observation of planar instability , 1995 .

[35]  Neville H Fletcher,et al.  Nonlinearity and chaos in acoustics , 1989 .

[36]  Kimihiko Yasuda,et al.  Nonlinear Forced Oscillations of a Shallow Spherical Shell , 1984 .

[37]  J. Laroche.  The use of the matrix pencil method for the spectrum analysis of musical signals , 1993 .

[38]  Alexander F. Vakakis,et al.  NON-LINEAR NORMAL MODES (NNMs) AND THEIR APPLICATIONS IN VIBRATION THEORY: AN OVERVIEW , 1997 .

[39]  A Chaigne,et al.  Time-domain simulation of damped impacted plates. II. Numerical model and results. , 2001, The Journal of the Acoustical Society of America.

[40]  S. A. Tobias,et al.  The Influence of Dynamical Imperfection on the Vibration of Rotating Disks , 1957 .

[41]  Cyril Touzé,et al.  ASYMMETRIC NON-LINEAR FORCED VIBRATIONS OF FREE-EDGE CIRCULAR PLATES. PART 1: THEORY , 2002 .

[42]  A. Chaigne,et al.  Time-domain simulation of damped impacted plates. I. Theory and experiments. , 2001, The Journal of the Acoustical Society of America.

[43]  Charles M. Krousgrill,et al.  NON-LINEAR DYNAMIC RESPONSE OF SHALLOW ARCHES TO HARMONIC FORCING , 1996 .

[44]  Arvind Raman,et al.  Effects of imperfection on the non-linear oscillations of circular plates spinning near critical speed , 2001 .

[45]  Wanda Szemplińska-Stupnicka,et al.  Bifurcations of harmonic solution leading to chaotic motion in the softening type Duffing's oscillator , 1988 .

[46]  F. Takens,et al.  On the nature of turbulence , 1971 .

[47]  J. G. Eisley,et al.  A Multiple Degree-of-Freedom Approach to Nonlinear Beam Vibrations , 1970 .

[48]  Pierre Schaeffer.  Traité des objets musicaux , 1966 .

[49]  Ali H. Nayfeh,et al.  Bifurcations in a forced softening duffing oscillator , 1989 .

[50]  P. R. Sethna,et al.  Local and global bifurcations in parametrically excited vibrations of nearly square plates , 1991 .

[51]  Joseph P. Cusumano,et al.  Chaotic non-planar vibrations of the thin elastica: Part II: Derivation and analysis of a low-dimensional model , 1995 .

[52]  S. A. Tobias.  Free Undamped Non-Linear Vibrations of Imperfect Circular Disks , 1957 .

[53]  O. Millet,et al.  Analyse dimensionnelle de l'équation de Navier et application à la théorie des plaques minces , 1997 .

[54]  J. Bendat,et al.  Random Data: Analysis and Measurement Procedures , 1987 .

[55]  Vimal Singh,et al.  Perturbation methods , 1991 .

[56]  N. Fletcher.  Nonlinear frequency shifts in quasispherical‐cap shells: Pitch glide in Chinese gongs , 1985 .

[57]  Antoine Chaigne,et al.  TIME-DOMAIN MODELING AND NUMERICAL SIMULATION OF A KETTLEDRUM , 1999 .

[58]  R. Benamar,et al.  The Effects Of Large Vibration Amplitudes On The Mode Shapes And Natural Frequencies Of Thin Elastic Structures, Part III: Fully Clamped Rectangular Isotropic Plates—Measurements Of The Mode Shape Amplitude Dependence And The Spatial Distribution Of Harmonic Distortion , 1994 .

[59]  C.J.H. Williams,et al.  The stability of nodal patterns in disk vibration , 1966 .

[60]  Jean Salençon,et al.  Mécanique des milieux continus , 2002 .

[61]  N. Fletcher,et al.  Music Producers.(Book Reviews: The Physics of Musical Instruments.) , 1991 .

[62]  E. Reissner.  ON THE THEORY OF BENDING OF ELASTIC PLATES , 1944 .

[63]  C. Vendhan.  A study of Berger equations applied to non-linear vibrations of elastic plates , 1975 .

[64]  M. Sathyamoorthy.  Nonlinear Vibrations of Plates: An Update of Recent Research Developments , 1996 .

[65]  R. White,et al.  The effects of large vibration amplitudes on the fundamental mode shape of a clamped-clamped uniform beam , 1984 .

[66]  R. Schmidt.  A current trend in shell theory: Constrained geometrically nonlinear Kirchhoff-Love type theories based on polar decomposition of strains and rotations , 1985 .

[67]  A. W. Leissa,et al.  Literature Review : Plate Vibration Research, 1976 - 1980: Complicating Effects , 1981 .

[68]  O. Millet,et al.  Justification du modèle bidimensionnel non linéaire de plaque par développement asymptotique des équations d'équilibre , 1997 .

[69]  Neville H Fletcher.  Nonlinear dynamics and chaos in musical instruments , 1993 .

[70]  H. Saunders,et al.  Acoustics: An Introduction to Its Physical Principles and Applications , 1984 .

[71]  Ali H. Nayfeh,et al.  On Nonlinear Modes of Continuous Systems , 1994 .

[72]  M. Sathyamoorthy,et al.  Nonlinear Vibration Analysis of Plates: A Review and Survey of Current Developments , 1987 .

[73]  A. W. Leissa,et al.  Recent Research in Plate Vibrations. 1973 - 1976: Complicating Effects , 1978 .

[74]  M. Sathyamoorthy,et al.  NONLINEAR VIBRATIONS OF PLATES - A REVIEW. , 1983 .

[75]  A. Nayfeh,et al.  Non-linear analysis of vibrations of irregular plates , 1977 .

[76]  André Gribenski,et al.  Acoustique et musique , 1973 .

[77]  A. W. Leissa,et al.  Recent studies in plate vibrations: 1981-85. I: Classical theory , 1987 .

[78]  Albert C. J. Luo,et al.  A quantitative stability and bifurcation analyses of the generalized duffing oscillator with strong nonlinearity , 1997 .

[79]  Charles M. Krousgrill,et al.  HARMONIC BALANCE AND CONTINUATION TECHNIQUES IN THE DYNAMIC ANALYSIS OF DUFFING'S EQUATION , 1997 .

[80]  Cyril Touzé,et al.  Lyapunov exponents from experimental time series. Application to cymbal vibrations , 2000 .

[81]  Philippe G. Ciarlet,et al.  A justification of the von Kármán equations , 1980 .

[82]  G. Duvaut.  Mécanique des milieux continus , 1990 .

[83]  H. Berger.  A new approach to the analysis of large deflections of plates , 1954 .

[84]  D. S. Dugdale.  Non-linear vibration of a centrally clamped rotating disc , 1979 .

[85]  John Guckenheimer,et al.  A Dynamical System Toolkit with an Interactive Graphical Interface , 1997 .

[86]  T. Rossing,et al.  Determination of modal coupling in vibrating rectangular plates , 1984 .