Numerical simulation of finite dimensional multibody nonsmooth mechanical systems

This review article focuses on the problems related to numerical simulation of finite dimensional nonsmooth multibody mechanical systems. The rigid body dynamical case is examined here. This class of systems involves complementarity conditions and impact phenomena, which make its study and numerical analysis a difficult problem that cannot be solved by relying on known Ordinary Differential Equation (ODE) or Differential Algebraic Equation (DAE) integrators only. The main techniques, mathematical tools, and existing algorithms are reviewed. The article utilizes 233 references.

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On the Determination of Joint Reactions in Multibody Mechanisms

Journal of Mechanical Design ◽

10.1115/1.1667944 ◽

2004 ◽

Vol 126 (2) ◽

pp. 341-350 ◽

Cited By ~ 24

Author(s):

Wojciech Blajer

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Algebraic Equation ◽

Matrix Inversion ◽

Differential Algebraic Equation ◽

Joint Coordinates ◽

Reduced Dimension ◽

Absolute Coordinates ◽

Coordinate Partitioning

In this paper some existing codes for the determination of joint reactions in multibody mechanisms are first reviewed. The codes relate to the DAE (differential-algebraic equation) dynamics formulations in absolute coordinates and in relative joint coordinates, and to the ODE (ordinary differential equation) formulations obtained by applying the coordinate partitioning method to these both coordinate types. On this background a novel efficient approach to the determination of joint reactions is presented, naturally associated with the reduced-dimension formulations of mechanism dynamics. By introducing open-constraint coordinates to specify the prohibited relative motions in the joints, pseudoinverse matrices to the constraint Jacobian matrices are derived in an automatic way. The involvement of the pseudo-inverses leads to schemes in which the joint reactions are obtained directly in resolved forms—no matrix inversion is needed as it is required in the classical codes. This makes the developed schemes especially well suited for both symbolic manipulators and computer implementations. Illustrative examples are provided.

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Large diffusivity finite-dimensional asymptotic behaviour of a semilinear wave equation

Journal of Applied Mathematics ◽

10.1155/s1110757x03212067 ◽

2003 ◽

Vol 2003 (8) ◽

pp. 409-427 ◽

Cited By ~ 2

Author(s):

Robert Willie

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Wave Equation ◽

Asymptotic Behaviour ◽

Nonlinear Term ◽

Damped Wave Equation ◽

Order Ordinary Differential Equation ◽

Finite Dimensional ◽

Type Boundary ◽

Large Diffusion

We study the effects of large diffusivity in all parts of the domain in a linearly damped wave equation subject to standard zero Robin-type boundary conditions. In the linear case, we show in a given sense that the asymptotic behaviour of solutions verifies a second-order ordinary differential equation. In the semilinear case, under suitable dissipative assumptions on the nonlinear term, we prove the existence of a global attractor for fixed diffusion and that the limiting attractor for large diffusion is finite dimensional.

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THE NONLINEARIZATION OF A (2+1)-DIMENSIONAL SOLITON EQUATION

Modern Physics Letters B ◽

10.1142/s0217984908017709 ◽

2008 ◽

Vol 22 (32) ◽

pp. 3179-3194

Author(s):

QIANG LIU ◽

DIAN-LOU DU

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Hamiltonian System ◽

Eigenvalue Problem ◽

Hamiltonian Systems ◽

Soliton Equation ◽

Completely Integrable ◽

Finite Dimensional ◽

Dimensional Soliton

Based on a 2 × 2 eigenvalue problem, a new (2+1)-dimensional soliton equation is proposed. Moreover, we obtain a finite-dimensional Hamiltonian system. Then we verify it is completely integrable in the Liouville sense. In the end, we introduce a set of Hk polynomial integrable, by which we can separate the solition equation into three compantiable Hamiltonian systems of ordinary differential equation.

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Numerical Simulation of the Evolution of Ice Covers Using the Scandinavian and Laurentide Ice Sheets as Examples

Journal of Glaciology ◽

10.1017/s002214300001162x ◽

1982 ◽

Vol 28 (99) ◽

pp. 267-272 ◽

Cited By ~ 1

Author(s):

D. D. Kvasov ◽

M. Ya. Verbitsky

Keyword(s):

Differential Equation ◽

Numerical Simulation ◽

Ordinary Differential Equation ◽

Mass Balance ◽

Climate Models ◽

Ice Sheets ◽

Ice Sheet ◽

Linear Parabolic Equation ◽

Last Glaciation ◽

Horizontal Size

AbstractA non-linear parabolic equation describing the evolution of an isothermal linearly viscous ice sheet is numerically solved in non-dimensional coordinates obtained by normalization over the horizontal size of a glacier. The horizontal size of the ice sheet is defined from the solution of an ordinary differential equation, the integral mass balance. For simple climate models, approximate relations describing the evolution of glaciers are proposed. These relations and palaeogeographical data are used to estimate changes in the mass balance on the surface of the Scandinavian and Laurentide ice sheets during retreat of the last glaciation.

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Numerical Simulation of the Evolution of Ice Covers Using the Scandinavian and Laurentide Ice Sheets as Examples

Journal of Glaciology ◽

10.3189/s002214300001162x ◽

1982 ◽

Vol 28 (99) ◽

pp. 267-272

Author(s):

D. D. Kvasov ◽

M. Ya. Verbitsky

Keyword(s):

Differential Equation ◽

Numerical Simulation ◽

Ordinary Differential Equation ◽

Mass Balance ◽

Climate Models ◽

Ice Sheets ◽

Ice Sheet ◽

Linear Parabolic Equation ◽

Last Glaciation ◽

Horizontal Size

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An Iterative Numerical Method for Solving the Lane–Emden Initial and Boundary Value Problems

International Journal of Computational Methods ◽

10.1142/s0219876218500202 ◽

2018 ◽

Vol 15 (04) ◽

pp. 1850020 ◽

Cited By ~ 3

Author(s):

Mohamed Ben-Romdhane ◽

Helmi Temimi

Keyword(s):

Differential Equation ◽

Numerical Simulation ◽

Ordinary Differential Equation ◽

Iterative Scheme ◽

Computational Cost ◽

Newton Raphson ◽

Previous Iteration ◽

Low Computational Cost ◽

Unknown Solution ◽

Quasilinearization Technique

In this paper, we propose fast iterative methods based on the Newton–Raphson–Kantorovich approximation in function space [Bellman and Kalaba, (1965)] to solve three kinds of the Lane–Emden type problems. First, a reformulation of the problem is performed using a quasilinearization technique which leads to an iterative scheme. Such scheme consists in an ordinary differential equation that uses the approximate solution from the previous iteration to yield the unknown solution of the current iteration. At every iteration, a further discretization of the problem is achieved which provides the numerical solution with low computational cost. Numerical simulation shows the accuracy as well as the efficiency of the method.

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