Octávio Inácio

Design of duct cross sectional areas in bass-trapping resonators for control rooms

Small rooms, such as the ones specifically designed for listening to amplified music, like control rooms in recording studios, face the problem of low-frequency over-enhancement by acoustic resonances. Several devices have been developed to tackle this problem, such as Helmoltz resonators. The number of controlled acoustic modes depends on several factors among which are the central frequency chosen, the modal density in that frequency range, and the coupling between the resonator and the room. In this paper we suggest that the efficiency of such resonators may be significantly improved if, instead of using basic Helmholtz or devices with uniform cross-section, more complex shape-optimized resonators are used, in order to cope with a larger number of undesirable acoustic modes. We apply optimization techniques to the uncoupled resonator, developed in our previous work, in order to obtain the optimal shapes for devices that resonate at a design set of acoustic eigenvalues, within imposed physical and/or geometrical constraints. One-dimensional and three-dimensional finite element models were implemented. The one-dimensional model was coupled to optimization techniques in order to achieve the design goal. We illustrate the proposed approach with two examples of resonator shapes and different design sets of absorption frequencies.

Simulation of the Oscillation Regimes of Bowed Bars: A Nonlinear Modal Approach

It is still a challenge to properly simulate the complex stick-slip behavior of multi-degree-of-freedom systems. In the present paper we investigate the self-excited nonlinear responses of bowed bars, using a time-domain modal approach, coupled with an explicit model for the frictional forces, which is able to emulate stick-slip behavior. This computational approach can provide very detailed simulations and is well suited to deal with systems presenting a dispersive behavior. The effects of the bar supporting fixture are included in the model, as well as a velocity-dependent friction coefficient. We present the results of numerical simulations, for representative ranges of the bowing velocity and normal force. Computations have been performed for constant-section aluminum bars, as well as for real vibraphone bars, which display a central undercutting, intended to help tuning the first modes. Our results show limiting values for the normal force FN and bowing velocity ybow , for which the “musical” self-sustained solutions exist. Beyond this “playability space”, double period and even chaotic regimes were found for specific ranges of the input parameters FN and ybow . As also displayed by bowed strings, the vibration amplitudes of bowed bars also increase with the bow velocity. However, in contrast to string instruments, bowed bars “slip” during most of the motion cycle. Another important difference is that, in bowed bars, the self-excited motions are dominated by the system first mode. Our numerical results are qualitatively supported by preliminary experimental results.Copyright

Linear modal stability analysis of bowed-strings

Linearised models are often invoked as a starting point to study complex dynamical systems. Besides their attractive mathematical simplicity, they have a central role for determining the stability properties of static or dynamical states, and can often shed light on the influence of the control parameters on the system dynamical behaviour. While the bowed string dynamics has been thoroughly studied from a number of points of view, mainly by time-domain computer simulations, this paper proposes to explore its dynamical behaviour adopting a linear framework, linearising the friction force near an equilibrium state in steady sliding conditions, and using a modal representation of the string dynamics. Starting from the simplest idealisation of the friction force given by Coulombs law with a velocity-dependent friction coefficient, the linearised modal equations of the bowed string are presented, and the dynamical changes of the system as a function of the bowing parameters are studied using linear stability analysis. From the computed complex eigenvalues and eigenvectors, several plots of the evolution of the modal frequencies, damping values, and modeshapes with the bowing parameters are produced, as well as stability charts for each system mode. By systematically exploring the influence of the parameters, this approach appears as a preliminary numerical characterisation of the bifurcations of the bowed string dynamics, with the advantage of being very simple compared to sophisticated numerical approaches which demand the regularisation of the nonlinear interaction force. To fix the idea about the potential of the proposed approach, the classic one-degree-of-freedom friction-excited oscillator is first considered, and then the case of the bowed string. Even if the actual stick-slip behaviour is rather far from the linear description adopted here, the results show that essential musical features of bowed string vibrations can be interpreted from this simple approach, at least qualitatively. Notably, the technique provides an instructive and original picture of bowed motions, in terms of groups of well-defined unstable modes, which is physically intuitive to discuss tonal changes observed in real bowed string.

An experimental modal analysis on the coimbra model of the Portuguese guitar

The most distinctive Portuguese traditional music style is Fado. In this form of Portuguese music, a singer is accompanied by two instruments: a classical guitar and a pear shaped plucked chordophone with six courses of double strings—the Portuguese guitar. There are two distinct types of this instrument—the Lisbon and the Coimbra models—named after the towns where the two different styles of Fado have developed. These guitars differ basically on their size and tuning, both comprising 6 orders of double steel strings, while the construction method (strutting patterns, wood species used, and soundboard thickness distribution) vary for different builders. As part of an ongoing research project that investigates the vibroacoustical behavior of this instrument for different types of designs, an experimental modal analysis of a fully assembled Coimbra guitar was performed. In this work, we present the results of this analysis showing the main characteristics of the frequency response curves and significant vibratory modes as compared to other similar plucked-string instruments.

Exploring the bowed string dynamical behavior using a linearized model approach

For several decades bowed-strings have captured the attention of many researchers aiming for a thorough understanding of this system. Different approaches have been adopted particularly in the time-domain numerical simulations of the self-excited nonlinear regimes. Recently, the authors have been exploring the advantages of using a linearized approach to this problem, with or without the body coupling influence. Despite the highly non-linear bow/string friction force, the problem can be linearized about the average sliding velocity, as usually done in break-squeal noise, and an eigenvalue analysis can offer interesting information. For example, this approach allowed exploring the modal dynamics of bowed-string/body coupled system, studying the prediction of modes instabilities and the possible emergence of a strongly coupled mode responsible for the wolf-note, among other features. Here, using the linearized modal dynamics of bowed-strings we look in detail to the behavior of the string modes as a functio...

Shape‐optimization of several multi‐modal resonators accounting for room/resonator acoustical coupling

Helmholtz resonators are often applied for the sound equalisation of control rooms, through adequate levelling of the low frequency acoustic modal room responses. In several recent papers we proposed to improve the eciency of such devices by, instead of using basic Helmholtz resonators with uniform cross-section, develop shape optimized multi-modal resonators in order to cope with a larger number of intrusive room modes. We thus showed the feasibility of resonator shape-optimization, in order to obtain a target set of acoustic eigenvalues, within imposed physical and/or geometrical constraints. More recently, we developed an ecient substructure theoretical approach to compute the coupled acoustical modes of rooms fitted with several multi-mode resonators, later also including viscous boundary layer absorption eects at the room/resonator interfaces. In the present paper we extend a further step these results by applying the previously developed optimization techniques to the fully coupled room/resonator model. We thus obtain truly representative results for the optimized complex acoustical problem, which highlight the potential of the proposed corrective methodology.

A theoretical dissipative analysis of optimized multimodal resonators coupled to room acoustics

Helmholtz resonators are often applied for the sound equalization of control rooms in recording studios, through adequate leveling of the low frequency acoustic modal room responses. The number of controlled acoustic modes depends on the central frequency and damping of resonators, as well as on the modal density of the controlled system within the resonator’s frequency range. In a recent paper, we proposed to improve the efficiency of such devices by, instead of using basic Helmholtz resonators, develop shape optimized multimodal resonators in order to cope with a larger number of intrusive room modes. In spite of the promising results thus obtained, further work is needed to demonstrate the feasibility of such an approach. The present paper is a further step in that direction by analyzing the acoustics of the fully coupled room/resonators system including dissapative effects. More specifically, using an acoustical substructure computational approach, we theoretically derive the coupled modes of rooms fi...

Linearized and nonlinear dynamics of bowed bars

Friction‐excited instruments have been for many decades an inexhaustible source of physical delight. In the last decades, spectacular improvements in computational power and in numerical methods enabled simulations of the self‐excited nonlinear regimes with considerable realism and detail. The authors of the present paper have achieved many such nonlinear simulations, using a powerful modal approach, and decided to investigate here in detail the characteristics of the linearized and nonlinear regimes of bowed bars. After stating the theoretical modeling approach for the nonlinear problem, we derive a corresponding linearized model, from which the complex eigenvalues and eigenvectors are computed as a function of the bowing parameters (friction parameters, normal force, and tangential velocity, accounting for the bowing location). We thus obtain plots of the modal frequencies, damping values, and complex mode shapes, as a function of the bowing parameters, as well as stability charts for each one of the sy...

Physical modeling of Tibetan bowls

Tibetan bowls produce rich penetrating sounds, used in musical contexts and to induce a state of relaxation for meditation or therapy purposes. To understand the dynamics of these instruments under impact and rubbing excitation, we developed a simulation method based on the modal approach, following our previous papers on physical modeling of plucked/bowed strings and impacted/bowed bars. This technique is based on a compact representation of the system dynamics, in terms of the unconstrained bowl modes. Nonlinear contact/friction interaction forces, between the exciter (puja) and the bowl, are computed at each time step and projected on the bowl modal basis, followed by step integration of the modal equations. We explore the behavior of two different‐sized bowls, for extensive ranges of excitation conditions (contact/friction parameters, normal force, and tangential puja velocity). Numerical results and experiments show that various self‐excited motions may arise depending on the playing conditions and, ...

Modeling the nonlinear string body coupled dynamics of bowed musical instruments

Most theoretical papers on bowed‐string instruments deal with isolated strings, pinned on fixed supports. In addition, the instrument body dynamics has not been accounted at all, or else by using extremely simplified models of the string/body interaction at the bridge. Such models have, nevertheless, been instrumental to the understanding of a very common and musically undesirable phenomenon—the ‘‘wolf note’’—a strong beating interplay between string and body vibrations. Cellos, bad and good, are prone to this problem. In previous work we developed a modal method to deal with friction‐excited strings, enabling effective simulations of such systems in which the strings were assumed decoupled from the instrument body. In the present paper our computational method is extended to incorporate the complex dynamics of real‐life instrument bodies, coupled to the string motions. In this approach, the string is coupled with experimental body data—body modes or impulse response at the bridge. Our computational method is illustrated through extensive parametric computations performed on a bowed cello. These numerical simulations show some light on interesting and less‐known features of wolf notes, in particular concerning the ranges of bowing‐parameters leading to their emergence, as well as the dependence of the beating frequency on the playing.