Energy conserving schemes for the simulation of musical instrument contact dynamics

Collisions are an innate part of the function of many musical instruments. Due to the nonlinear nature of contact forces, special care has to be taken in the construction of numerical schemes for simulation and sound synthesis. Finite difference schemes and other time-stepping algorithms used for musical instrument modelling purposes are normally arrived at by discretising a Newtonian description of the system. However because impact forces are non-analytic functions of the phase space variables, algorithm stability can rarely be established this way. This paper presents a systematic approach to deriving energy conserving schemes for frictionless impact modelling. The proposed numerical formulations follow from discretising Hamilton׳s equations of motion, generally leading to an implicit system of nonlinear equations that can be solved with Newton׳s method. The approach is first outlined for point mass collisions and then extended to distributed settings, such as vibrating strings and beams colliding with rigid obstacles. Stability and other relevant properties of the proposed approach are discussed and further demonstrated with simulation examples. The methodology is exemplified through a case study on tanpura string vibration, with the results confirming the main findings of previous studies on the role of the bridge in sound generation with this type of string instrument.

A Lagrangian approach for modeling road collisions using second-order models of traffic flow

In [M. Herty, A. Klein, S. Moutari, V. Schleper, and G. Steinaur, IMA J. Appl. Math., 78(5), 1087–1108, 2013] and [M. Herty and V. Schleper, ZAMM J. Appl. Math. Mech., 91, 763–776, 2011], a macroscopic approach, derived from fluid-dynamics models, has been introduced to infer traffic conditions prone to road traffic collisions along highways’ sections. In these studies, the governing equations are coupled within an Eulerian framework, which assumes fixed interfaces between the models. A coupling in Lagrangian coordinates would enable us to get rid of this (not very realistic) assumption. In this paper, we investigate the well-posedness and the suitability of the coupling of the governing equations within the Lagrangian framework. Further, we illustrate some features of the proposed approach through some numerical simulations.

Modeling relativistic soliton interactions in overdense plasmas: A perturbed nonlinear Schrodinger equation framework

We investigate the dynamics of localized solutions of the relativistic cold-fluid plasma model in the small but finite amplitude limit, for slightly overcritical plasma density. Adopting a multiple scale analysis, we derive a perturbed nonlinear Schrodinger equation that describes the evolution of the envelope of circularly polarized electromagnetic field. Retaining terms up to fifth order in the small perturbation parameter, we derive a self-consistent framework for the description of the plasma response in the presence of localized electromagnetic field. The formalism is applied to standing electromagnetic soliton interactions and the results are validated by simulations of the full cold-fluid model. To lowest order, a cubic nonlinear Schrodinger equation with a focusing nonlinearity is recovered. Classical quasiparticle theory is used to obtain analytical estimates for the collision time and minimum distance of approach between solitons. For larger soliton amplitudes the inclusion of the fifth-order terms is essential for a qualitatively correct description of soliton interactions. The defocusing quintic nonlinearity leads to inelastic soliton collisions, while bound states of solitons do not persist under perturbations in the initial phase or amplitude

Meanline Modeling of Inlet Recirculation in Automotive Turbocharger Centrifugal Compressors

This study provides a novel meanline modeling approach for centrifugal compressors. All compressors analyzed are of the automotive turbocharger variety and have typical upstream geometry with no casing treatments or preswirl vanes. Past experience dictates that inducer recirculation is prevalent toward surge in designs with high inlet shroud to outlet radius ratios; such designs are found in turbocharger compressors due to the demand for operating range. The aim of the paper is to provide further understanding of impeller inducer flow paths when operating with significant inducer recirculation. Using three-dimensional (3D) computational fluid dynamics (CFD) and a single-passage model, the flow coefficient at which the recirculating flow begins to develop and the rate at which it grows are used to assess and correlate work and angular momentum delivered to the incoming flow. All numerical modeling has been fully validated using measurements taken from hot gas stand tests for all compressor stages. The new modeling approach links the inlet recirculating flow and the pressure ratio characteristic of the compressor. Typically for a fixed rotational speed, between choke and the onset of impeller inlet recirculation the pressure ratio rises gradually at a rate dominated by the aerodynamic losses. However, in modern automotive turbocharger compressors where operating range is paramount, the pressure ratio no longer changes significantly between the onset of recirculation and surge. Instead the pressure ratio remains relatively constant for reducing mass flow rates until surge occurs. Existing meanline modeling techniques predict that the pressure ratio continues to gradually rise toward surge, which when compared to test data is not accurate. A new meanline method is presented here which tackles this issue by modeling the direct effects of the recirculation. The result is a meanline model that better represents the actual fluid flow seen in the CFD results and more accurately predicts the pressure ratio and efficiency characteristics in the region of the compressor map affected by inlet recirculation.

Data-Driven Dynamic Modeling of Coupled Thermal and Electric Outputs of Microturbines

Microturbines are among the most successfully commercialized distributed energy resources, especially when they are used for combined heat and power generation. However, the interrelated thermal and electrical system dynamic behaviors have not been fully investigated. This is technically challenging due to the complex thermo-fluid-mechanical energy conversion processes which introduce multiple time-scale dynamics and strong nonlinearity into the analysis. To tackle this problem, this paper proposes a simplified model which can predict the coupled thermal and electric output dynamics of microturbines. Considering the time-scale difference of various dynamic processes occuring within microturbines, the electromechanical subsystem is treated as a fast quasi-linear process while the thermo-mechanical subsystem is treated as a slow process with high nonlinearity. A three-stage subspace identification method is utilized to capture the dominant dynamics and predict the electric power output. For the thermo-mechanical process, a radial basis function model trained by the particle swarm optimization method is employed to handle the strong nonlinear characteristics. Experimental tests on a Capstone C30 microturbine show that the proposed modeling method can well capture the system dynamics and produce a good prediction of the coupled thermal and electric outputs in various operating modes.