Simulation of dynamic pinion course using Runge-Kutta's method and impact modeling

Once defined the relationship between the Starter Motor components and their functions, it is possible to develop a mathematical model capable to predict the Starter behavior during operation. One important aspect is the engagement system behavior. The development of a mathematical tool capable of predicting it is a valuable step in order to reduce the design time, cost and engineering efforts. A mathematical model, represented by differential equations, can be developed using physics laws, evaluating force balance and energy flow through the systems degrees of freedom. Another important physical aspect to be considered in this modeling is the impact conditions (particularly on the pinion and ring-gear contact). This work is a report of those equations application on available mathematical software and the resolution of those equations by Runge-Kutta's numerical integration method, in order to build an accessible engineering tool. Copyright © 2011 SAE International.

Uma abordagem para problemas e controle ótimo via métodos de Runge-Kutta e análise de erro

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Lobatto deferred correction for stiff two-point boundary value problems

An iterated deferred correction algorithm based on Lobatto Runge-Kutta formulae is developed for the efficient numerical solution of nonlinear stiff two-point boundary value problems. An analysis of the stability properties of general deferred correction schemes which are based on implicit Runge-Kutta methods is given and results which are analogous to those obtained for initial value problems are derived. A revised definition of symmetry is presented and this ensures that each deferred correction produces an optimal increase in order. Finally, some numerical results are given to demonstrate the superior performance of Lobatto formulae compared with mono-implicit formulae on stiff two-point boundary value problems. (C) 1998 Elsevier B.V. Ltd. All rights reserved.

Critical number of atoms for attractive Bose-Einstein condensates with cylindrically symmetrical traps

The critical number of atoms for Bose-Einstein condensates with cylindrically symmetrical traps were calculated. The time evolution of the condensate was also studied at changing ground state. A conjecture on higher-order nonlinear effects was also discussed to determine its signal and strength. The results show that by exchanging frequencies, the geometry favors the condensation of larger number of particles.

Object oriented modeling of piston engines

This work reports a conception phase of a piston engine global model. The model objective is forecast the motor performance (power, torque and specific consumption as a function of rotation and environmental conditions). Global model or Zero-dimensional is based on flux balance through each engine component. The resulting differential equations represents a compressive unsteady flow, in which, all dimensional variables are areas or volumes. A review is presented first. The ordinary differential equation system is presented and a Runge-Kutta method is proposed to solve it numerically. The model includes the momentum conservation equation to link the gas dynamics with the engine moving parts rigid body mechanics. As an oriented to objects model the documentation follows the UML standard. A discussion about the class diagrams is presented, relating the classes with physical model related. The OOP approach allows evolution from simple models to most complex ones without total code rewrite. Copyright © 2001 Society of Automotive Engineers, Inc.

Numerical assessment of mass conservation on a MAC-type method for viscoelastic free surface flows

A numerical study of mass conservation of MAC-type methods is presented, for viscoelastic free-surface flows. We use an implicit formulation which allows for greater time steps, and therefore time marching schemes for advecting the free surface marker particles have to be accurate in order to preserve the good mass conservation properties of this methodology. We then present an improvement by using a Runge-Kutta scheme coupled with a local linear extrapolation on the free surface. A thorough study of the viscoelastic impacting drop problem, for both Oldroyd-B and XPP fluid models, is presented, investigating the influence of timestep, grid spacing and other model parameters to the overall mass conservation of the method. Furthermore, an unsteady fountain flow is also simulated to illustrate the low mass conservation error obtained.

Three-dimensional transient complex free surface flows: Numerical simulation of XPP fluid

In this paper we present a finite difference MAC-type approach for solving three-dimensional viscoelastic incompressible free surface flows governed by the eXtended Pom-Pom (XPP) model, considering a wide range of parameters. The numerical formulation presented in this work is an extension to three-dimensions of our implicit technique [Journal of Non-Newtonian Fluid Mechanics 166 (2011) 165-179] for solving two-dimensional viscoelastic free surface flows. To enhance the stability of the numerical method, we employ a combination of the projection method with an implicit technique for treating the pressure on the free surfaces. The differential constitutive equation of the fluid is solved using a second-order Runge-Kutta scheme. The numerical technique is validated by performing a mesh refinement study on a pipe flow, and the numerical results presented include the simulation of two complex viscoelastic free surface flows: extrudate-swell problem and jet buckling phenomenon. © 2013 Elsevier B.V.

Representation of transmission lines: comparison of models and parameters distributed discrete parameters

A transmission line is characterized by the fact that its parameters are distributed along its length. This fact makes the voltages and currents along the line to behave like waves and these are described by differential equations. In general, the differential equations mentioned are difficult to solve in the time domain, due to the convolution integral, but in the frequency domain these equations become simpler and their solutions are known. The transmission line can be represented by a cascade of π circuits. This model has the advantage of being developed directly in the time domain, but there is a need to apply numerical integration methods. In this work a comparison of the model that considers the fact that the parameters are distributed (Universal Line Model) and the fact that the parameters considered concentrated along the line (π circuit model) using the trapezoidal integration method, and Simpson's rule Runge-Kutta in a single-phase transmission line length of 100 km subjected to an operation power. © 2003-2012 IEEE.

Bose-Einstein condensation dynamics in three dimensions by the pseudospectral and finite-difference methods

We suggest a pseudospectral method for solving the three-dimensional time-dependent Gross-Pitaevskii (GP) equation, and use it to study the resonance dynamics of a trapped Bose-Einstein condensate induced by a periodic variation in the atomic scattering length. When the frequency of oscillation of the scattering length is an even multiple of one of the trapping frequencies along the x, y or z direction, the corresponding size of the condensate executes resonant oscillation. Using the concept of the differentiation matrix, the partial-differential GP equation is reduced to a set of coupled ordinary differential equations, which is solved by a fourth-order adaptive step-size control Runge-Kutta method. The pseudospectral method is contrasted with the finite-difference method for the same problem, where the time evolution is performed by the Crank-Nicholson algorithm. The latter method is illustrated to be more suitable for a three-dimensional standing-wave optical-lattice trapping potential.

Integration Methods Used in Numerical Simulations of Electromagnetic Transients

This paper made an analysis of some numerical integration methods that can be used in electromagnetic transient simulations. Among the existing methods, we analyzed the trapezoidal integration method (or Heun formula), Simpson's Rule and Runge-Kutta. These methods were used in simulations of electromagnetic transients in power systems, resulting from switching operations and maneuvers that occur in transmission lines. Analyzed the characteristics such as accuracy, computation time and robustness of the methods of integration.

Modelo dinâmico da coluna lombar humana, com solicitação de esforço postero-anterior: análise com rigidez viscoelástica não linear

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Métodos numéricos para equações diferencias ordinárias

Pós-graduação em Matemática Universitária - IGCE

Estudos de técnicas de reinicialização em métodos level-set e suas aplicações em escoamentos bifásicos

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Iterated deferred correction for linear two-point boundary value problems

An analysis of iterated deferred correction based on various classes of implicit Runge-Kutta formulae is given. Out of different possibilities considered, it is shown that those based purely on Lobatto formulae have the best stability. The enhanced stability of Lobatto schemes is very important for the efficient integration of excessively stiff boundary value problems and this is demonstrated by means of some numerical results.

Tm and Tm-Tb-doped germanate glasses for S-band amplifiers

The mechanism involved in the Tm(3+)((3)F(4)) -> Tb(3+)((7)F(0,1,2)) energy transfer as a function of the Tb concentration was investigated in Tm:Tb-doped germanate (GLKZ) glass. The experimental transfer rate was determined from the best fit of the (3)F(4) luminescence decay due to the Tm -> Tb energy transfer using the Burshtein model. The result showed that the 1700 nm emission from (3)F(4) can be completely quenched by 0.8 mol% of Tb(3+). As a consequence, the (7)F(3) state of Tb(3+) interacts with the (3)H(4) upper excited state of TM(3+) slighting decreasing its population. The effective amplification coefficient beta(cm(-1)) that depends on the population density difference Delta n = n((3)H(4))-n((3)F(4)) involved in the optical transition of Tm(3+) (S-band) was calculated by solving the rate equations of the system for continuous pumping with laser at 792 nm, using the Runge-Kutta numerical method including terms of fourth order. The population density inversion An as a function of Tb(3+) concentration was calculated by computational simulation for three pumping intensities, 0.2, 2.2 and 4.4 kWcm(-2). These calculations were performed using the experimental Tm -> Tb transfer rates and the optical constants of the Tm (0.1 mol%) system. It was demonstrated that 0.2 mol% of Tb(3+) propitiates best population density inversion of Tin(3+) maximizing the amplification coefficient of Tm-doped (0.1 mol%) GLKZ glass when operating as laser intensity amplification at 1.47 mu m. (C) 2007 Elsevier B.V. All rights reserved.