SOUND SYNTHESIS OF GONGS OBTAINED FROM NONLINEAR THIN PLATES VIBRATIONS: COMPARISON BETWEEN A MODAL APPROACH AND A FINITE DIFFERENCE SCHEME

International audienceThe sound of a gong is simulated through the vibrations of thin elastic plates. The dynamical equations are necessarily nonlinear, crashing and shimmering being typical nonlinear effects. In this work two methods are used to simulate the nonlinear plates: a finite difference scheme and a modal approach. The striking force is approximated to the first order by a raised cosine of varying amplitude and contact duration acting on one point of the surface. It will be seen that for linear and moderately nonlinear vibrations the modal approach is particularly appealing as it allows the implementation of a rich damping mechanism by introducing a damping coefficient for each mode. In this way, the frequency-dependent decay rates can be tuned to get a very realistic sound. However, in many cases cymbal vibrations are found in strongly nonlinear regimes, where an energy cascade through lengthscales brings energy up to high-frequency modes. Hence, the number of modes retained in the truncation becomes a crucial parameter of the simulation. In this sense the finite difference scheme is usually better suited for reproducing crash and gong-like sounds, because this scheme retains all the modes up to (almost) Nyquist. However, the modal equations will be shown to have useful symmetry properties that can be used to speed up the off-line calculation process, leading to large memory and time savings and thus giving the possibility to simulate higher frequency ranges using modes

Interim report on Media Analysis

PACHELBEL WP4 “Stimulus Materials” uses findings from WP3 (Policy Assumptions) and from additional sources to prepare stimulus materials for the group-based process to be implemented in WP5. The output, informed by the present report, will be a set of materials to inform and stimulate the group-based process. These will take the form of real or simulated media coverage and/or documentary materials produced by various sources, scenarios, vignettes, and dramatised accounts. Another output, also informed by this report, will be an individual questionnaire for use in the group-based process. The present deliverable is centred on one of the data-gathering and analytic activities set up by WP4 to identify pertinent representational elements that should be included in the future stimulus materials, country by country. “Representational elements” have been defined in WP4 as typical images, anecdotes, examples, and references which are used by policy actors to explain and justify policy choices within the policy domains pertinent to PACHELBEL. Particular attention is given to references made to citizens, their perceptions and behaviours. In Task 4.2, PACHELBEL partners gathered representational elements in their respective contexts. To support this task, a “media analysis” template was developed by WPL SYMLOG for discussion at the second Consortium project meeting (Dorking, Mo. 6). Criteria were agreed for the analysis of a selection of actual publications in a range of media (print periodicals, public information materials disseminated by authorities, etc.). In Summer 2010, partners in each country used the template to analyze and report a sample of several dozen articles in selected policy areas. This interim report (D4.2) recalls methodology (Part 1), presents representational elements country by country (Part 2) and provides a summary overview of similarities and contrasts across country samples (Part 3). Conclusions and next steps are presented in Part 4. Also provided are a simplified media analysis template (Annex 1) and the compiled basic frequency analysis (Annex 2)

Non-iterative simulation methods for virtual analog modelling

The simulation of nonlinear components is central to virtual analog simulation. In audio effects, circuits often include devices such as diodes and transistors, mostly operating in a strongly nonlinear regime. Mathematical models are in the form of systems of nonlinear ordinary differential equations (ODEs), and traditional integrators, such as the trapezoid and midpoint methods, can be employed as solvers. These methods are fully implicit and require the solution of a nonlinear algebraic system at each time step, introducing further complications regarding the existence and uniqueness of the solution, as well as the choice of halting conditions for the iterative root finder. On the other hand, fast explicit methods such as Forward Euler, are prone to unstable behaviour at standard audio sample rates. For these reasons, in this work, a family of linearly-implicit schemes is presented. These schemes take the form of a perturbation expansion, making the construction of higher-order schemes possible. Compared with classic implicit designs, the proposed methods have the advantage of efficiency, since the update is computed in a single iteration. Furthermore, the existence and uniqueness of the update are proven by simple inspection of the update matrix. Compared to classic explicit designs, the proposed schemes display stable behaviour at standard audio sample rates. In the case of a single scalar ODE, sufficient conditions for numerical stability can be derived, imposing constraints on the choice of the sampling rate. Several theoretical results are provided, as well as numerical examples for typical stiff equations used in virtual analog modelling

Fast explicit algorithms for Hamiltonian numerical integration

Numerical integration methods for Hamiltonian systems are of importance across many disciplines, including musical acoustics, where many systems of interest are very nearly lossless. Of particular interest are methods possessing a conserved pseu-doenergy. Though most such methods have an implicit character, an explicit method was proposed recently by Marazzato et al. The proposed method relies on a continuous integration which must be performed exactly in order for the conservation property to hold-as a result, it holds only approximately under numerical quadrature. Here, we show an explicit scheme for Hamiltonian integration, with a different choice of pseudoenergy, which is exactly conserved. Most importantly, a fast implementation is possible through the use of structured matrix inversion, and in particular Sherman Morrison inversion of the rank 1 perturbation of a matrix. Applications to the cases of fully nonlinear string vibration, and to the Föppl-von Kármán system describing large amplitude plate vibration are illustrated. Computation times are on par with the simplest non-conservative methods, such as Störmer integration

SIMULATION OF THE SNARE-MEMBRANE COLLISION IN MODAL FORM USING THE SCALAR AUXILIARY VARIABLE (SAV) METHOD

Collisions play an essential role in the sound production of many musical instruments, such as in the snare drum. Here, collisions occur between the stick and the batter head and between the snares and the bottom head. The latter involve interactions between fully distributed objects and are the subject of this work. From a simulation standpoint, simple explicit or semi-implicit schemes are prone to unstable numerical behaviour and an appropriate energy-conserving framework is required for stable simulation designs. Usually, this is accomplished via fully-implicit designs that are known to conserve energy but that require iterative solvers such as Newton-Raphson. Other than representing a computational bottleneck, iterative schemes present a variable operational cost per timestep and, furthermore, are serial in nature. This work will explore the possibility of simulating the snare-membrane collision using explicit designs obtained via a quadratisation of the nonlinear potential energy. A modal function basis will be employed for the spatial discretisation, allowing for fine-tuning damping ratios and natural frequencies

Explicit exactly energy-conserving methods for Hamiltonian systems

For Hamiltonian systems, simulation algorithms that exactly conserve numerical energy or pseudo-energy have seen extensive investigation. Most available methods either require the iterative solution of nonlinear algebraic equations at each time step, or are explicit, but where the exact conservation property depends on the exact evaluation of an integral in continuous time. Under further restrictions, namely that the potential energy contribution to the Hamiltonian is non-negative, newer techniques based on invariant energy quadratisation allow for exact numerical energy conservation and yield linearly implicit updates, requiring only the solution of a linear system at each time step. In this article, it is shown that, for a general class of Hamiltonian systems, and under the non-negativity condition on potential energy, it is possible to arrive at a fully explicit method that exactly conserves numerical energy. Furthermore, such methods are unconditionally stable, and are of comparable computational cost to the very simplest integration methods (such as Störmer-Verlet). A variant of this scheme leading to a conditionally-stable method is also presented, and follows from a splitting of the potential energy. Various numerical results are presented, in the case of the classic test problem of Fermi, Pasta and Ulam and for nonlinear systems of partial differential equations, including those describing high amplitude vibration of strings and plates

Efficient simulation of the yaybahar using a modal approach

This work presents a physical model of the yaybahar, a recently invented acoustic instrument. Here, output from a bowed string is passed through a long spring, before being amplified and propagated in air via a membrane. The highly dispersive character of the spring is responsible for the typical synthetic tonal quality of this instrument. Building on previous literature, this work presents a modal discretisation of the full system, with fine control over frequency-dependent decay times, modal amplitudes and frequencies, all essential for an accurate simulation of the dispersive characteristics of reverberation. The string-bow-bridge system is also solved in the modal domain, using recently developed non-iterative numerical methods allowing for efficient simulation