Antoine Chaigne

Numerical simulations of piano strings. I. A physical model for a struck string using finite difference methods

The first attempt to generate musical sounds by solving the equations of vibrating strings by means of finite difference methods (FDM) was made by Hiller and Ruiz [J. Audio Eng. Soc. 19, 462–472 (1971)]. It is shown here how this numerical approach and the underlying physical model can be improved in order to simulate the motion of the piano string with a high degree of realism. Starting from the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, a numerical finite difference scheme is derived, from which the time histories of string displacement and velocity for each point of the string are computed in the time domain. The interacting force between hammer and string, as well as the force acting on the bridge, are given by the same scheme. The performance of the model is illustrated by a few examples of simulated string waveforms. A brief discussion of the aspects of numerical stability and dispersion with reference to the proper choice of sampling parameters is also included.

Numerical simulations of xylophones. I. Time-domain modeling of the vibrating bars

A time-domain modeling of xylophone bars excited by the blow of a mallet is presented. The flexural vibrations of the bar, with nonuniform cross section, are modeled by a one-dimensional Euler–Bernoulli equation, modified by the addition of two damping terms for the modeling of losses and a restoring force for the modeling of the stiffness of the suspending cord. The action of the mallet against the bar is described by Hertz’s law of contact for linear elastic bodies. This action appears as a force density term on the right-hand side of the bending wave equation. The model is completed by the equation of motion for the mallet, and by free–free boundary conditions for the bar. The bending wave equation of the bar is put into a numerical form by means of an implicit finite-difference scheme, which ensures a sufficient spatial resolution for an accurate tuning of the bar. The geometrical, elastic, and damping parameters of the model are derived from experiments carried out on actual xylophones and mallets. T...

The psychomechanics of simulated sound sources: Material properties of impacted bars

Sounds convey information about the materials composing an object. Stimuli were synthesized using a computer model of impacted plates that varied their material properties: viscoelastic and thermoelastic damping and wave velocity (related to elasticity and mass density). The range of damping properties represented a continuum between materials with predominant viscoelastic and thermoelastic damping (glass and aluminum, respectively). The perceptual structure of the sounds was inferred from multidimensional scaling of dissimilarity judgments and from their categorization as glass or aluminum. Dissimilarity ratings revealed dimensions that were closely related to mechanical properties: a wave-velocity-related dimension associated with pitch and a damping-related dimension associated with timbre and duration. When asked to categorize sounds, however, listeners ignored the cues related to wave velocity and focused on cues related to damping. In both dissimilarity-rating and identification experiments, the results were independent of the material of the mallet striking the plate (rubber or wood). Listeners thus appear to select acoustical information that is reliable for a given perceptual task. Because the frequency changes responsible for detecting changes in wave velocity can also be due to changes in geometry, they are not as reliable for material identification as are damping cues.

Time-domain simulation of damped impacted plates. I. Theory and experiments

A time-domain formulation for the flexural vibrations in damped rectangular isotropic and orthotropic plates is developed, in order to investigate transient excitation of plates by means of sound synthesis. The model includes three basic mechanisms of damping (thermoelasticity, viscoelasticity and radiation) using a general differential operator. The four rigidity factors of the plate are modified by perturbation terms, each term corresponding to one specific damping mechanism. The first damping term is derived from the coupling between the thermoelastic stress-strain relations and the heat diffusion equation. The second term is obtained from the general differential formulation of viscoelasticity. The third term is obtained through a Pade approximation of the damping factor which governs the coupling of the plate with the surrounding air. The decay factors predicted by the model reproduce adequately the dependence on both dimensions and frequency of the decay factors measured on rectangular plates of various sizes and thicknesses made of four different materials (aluminum, glass, carbon fiber, and wood). The numerical resolution of the complete problem, including initial and boundary conditions, and the comparison between real and simulated sounds are presented in a companion paper.

Time-domain modeling and numerical simulation of a kettledrum

A kettledrum is made of a circular elastic membrane stretched over an enclosed air cavity. It is set into vibration by the impact of the mallet. The motion of the membrane is coupled with both the external and internal sound field. A time-domain modeling of this instrument is presented which describes the motion of the mallet and its nonlinear interaction with the membrane, the transverse displacement of the membrane, and the sound pressure inside and outside the cavity. Based on a variational formulation of the problem, which uses the pressure jump over the boundaries of the instrument as a new variable, a numerical scheme is derived by means of three-dimensional finite elements. Higher-order absorbing conditions are used to simulate the free space. The validity of the model is illustrated by successive snapshots showing the pressure fields and the displacement of the membrane. In addition, time histories of energetic quantities help in explaining how the energy is balanced between mallet, membrane, and acoustic field in real instruments. Simulated external pressures show particularly good agreement with the sound field radiated by real instruments in both time and frequency domains.

Asymmetric non-linear forced vibrations of free-edge circular plates. Part II: experiments

This article is devoted to an experimental validation of a theoretical model presented in an earlier contribution by the same authors. The non-linear forced vibrations of circular plates, with the excitation frequency close to the natural frequency of an asymmetric mode, are investigated. The experimental set-up, which allows one to perform precise measurements of the vibration amplitudes of the two preferential configurations, is presented. Experimental resonance curves showing the amplitude and the phase of each configuration as functions of the driving frequency are compared to the theoretical ones, leading to a quantitative validation of the predictions given by the model. Finally, all the approximations used are systematically discussed, in order to show the scope and relevance of the approach.

Numerical simulations of piano strings. II. Comparisons with measurements and systematic exploration of some hammer‐string parameters

A physical model of the piano string, using finite difference methods, has recently been developed. [Chaigne and Askenfelt, J. Acoust. Soc. Am. 95, 1112–1118 (1994)]. The model is based on the fundamental equations of a damped, stiff string interacting with a nonlinear hammer, from which a numerical finite difference scheme is derived. In the present study, the performance of the model is evaluated by systematic comparisons between measured and simulated piano tones. After a verification of the accuracy of the method, the model is used as a tool for systematically exploring the influence of string stiffness, relative striking position, and hammer‐string mass ratio on string waveforms and spectra.

Time-domain simulation of damped impacted plates. II. Numerical model and results

A time-domain model for the flexural vibrations of damped plates was presented in a companion paper [Part I, J. Acoust. Soc. Am. 109, 1422-1432 (2001)]. In this paper (Part II), the damped-plate model is extended to impact excitation, using Hertzs law of contact, and is solved numerically in order to synthesize sounds. The numerical method is based on the use of a finite-difference scheme of second order in time and fourth order in space. As a consequence of the damping terms, the stability and dispersion properties of this scheme are modified, compared to the undamped case. The numerical model is used for the time-domain simulation of vibrations and sounds produced by impact on isotropic and orthotropic plates made of various materials (aluminum, glass, carbon fiber and wood). The efficiency of the method is validated by comparisons with analytical and experimental data. The sounds produced show a high degree of similarity with real sounds and allow a clear recognition of each constitutive material of the plate without ambiguity.

Numerical simulations of xylophones. II. Time-domain modeling of the resonator and of the radiated sound pressure

This paper presents a time-domain modeling for the sound pressure radiated by a xylophone and, more generally, by mallet percussion instruments such as the marimba and vibraphone, using finite difference methods. The time-domain model used for the one-dimensional (1-D) flexural vibrations of a nonuniform bar has been described in a previous paper by Chaigne and Doutaut [J. Acoust. Soc. Am. 101, 539–557 (1997)] and is now extended to the modeling of the sound-pressure field radiated by the bar coupled with a 1-D tubular resonator. The bar is viewed as a linear array of equivalent oscillating spheres. A fraction of the bar field excites the tubular resonator which, in turn, radiates sound with a certain delay. In the present model, the open end of the resonator is represented by an equivalent pulsating sphere. The total sound field is obtained by summing the respective contributions of the bar and tube. Particular care is given for defining a valid approximation of the radiation impedance, both in continuous and discrete time domain, on the basis of Kreiss’s theory. The model is successful in reproducing the main features of real instruments: sharp attack, tuning of the bar, directivity, tone color, and aftersound due to the bar-resonator coupling.

Modeling and simulation of a grand piano.

A time-domain global modeling of a grand piano is presented. The string model includes internal losses, stiffness, and geometrical nonlinearity. The hammer-string interaction is governed by a nonlinear dissipative compression force. The soundboard is modeled as a dissipative bidimensional orthotropic Reissner-Mindlin plate where the presence of ribs and bridges is treated as local heterogeneities. The coupling between strings and soundboard at the bridge allows the transmission of both transverse and longitudinal waves to the soundboard. The soundboard is coupled to the acoustic field, whereas all other parts of the structure are supposed to be perfectly rigid. The acoustic field is bounded artificially using perfectly matched layers. The discrete form of the equations is based on original energy preserving schemes. Artificial decoupling is achieved, through the use of Schur complements and Lagrange multipliers, so that each variable of the problem can be updated separately at each time step. The capability of the model is highlighted by series of simulations in the low, medium, and high register, and through comparisons with waveforms recorded on a Steinway D piano. Its ability to account for phantom partials and precursors, consecutive to string nonlinearity and inharmonicity, is particularly emphasized.