Energy and momentum conserving methods of arbitrary order for the numerical integration of equations of motion

1975 ◽  
Vol 25 (4) ◽  
pp. 323-346 ◽  
Author(s):  
Robert A. LaBudde ◽  
Donald Greenspan
Keyword(s):  
Numerical Integration ◽  
Arbitrary Order ◽  
Equations Of Motion ◽  
Energy And Momentum

Hertz Contact Force Model With Permanent Indentation in Impact Analysis of Solids

1992 ◽  
Author(s):  
Hamid M. Lankarani ◽  
Parviz E. Nikravesh
Keyword(s):  
Contact Force ◽  
Impact Analysis ◽  
Equations Of Motion ◽  
Solid Particles ◽  
Hertzian Contact ◽  
Contact Period ◽  
Force Model ◽  
Unknown Parameters ◽  
Contact Force Model ◽  
Energy And Momentum

Abstract A continuous analysis method for the direct-central impact of two solid particles is presented. Based on the assumption that local plasticity effects are the sole factor accounting for the dissipation of energy in impact, a Hertzian contact force model with permanent indentation is constructed. Utilizing energy and momentum considerations, the unknown parameters in the model are analytically evaluated in terms of a given coefficient of restitution and velocities before impact. The equations of motion of the two solids may then be integrated forward in time knowing the variation of the contact force during the contact period. For Illustration, an impact of two soft metallic particles is studied.

The DDA Method

2016 ◽  
pp. 90-102
Author(s):  
Katalin Bagi
Keyword(s):  
Numerical Integration ◽  
Displacement Field ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Discontinuous Deformation Analysis ◽  
Deformation Analysis ◽  
Short Overview ◽  
Abrupt Changes ◽  
Discontinuous Deformation ◽  
Iteration Technique

“DDA” stands for “Discontinuous Deformation Analysis”, suggesting that the displacement field of the analyzed domain shows abrupt changes on the element boundaries in the model. This chapter introduces the theoretical fundaments of DDA: mechanical characteristics of the elements together with the basic degrees of freedom, contact behavior, the equations of motion and their numerical integration with the help of Newmark's beta-method taking into account contact creation, loss and sliding with the help of an open-close iteration technique. Finally, a short overview on practical and scientific applications for masonry structures is given.

scholarly journals Treatment of Close Approaches in the Numerical Integration of the Gravitational Problem of N Bodies

1971 ◽  
Vol 10 ◽  
pp. 133-150 ◽  
Author(s):  
D. G. Bettis ◽  
V. Szebehely
Keyword(s):  
Numerical Integration ◽  
Body Problem ◽  
Considerable Increase ◽  
Equations Of Motion ◽  
Computer Time ◽  
Tidal Effects ◽  
Basic Ideas ◽  
Gravitational Problem ◽  
Considerable Detail ◽  
Numerical Approaches

AbstractOne of the main difficulties encountered in the numerical integration of the gravitational n-body problem is associated with close approaches. The singularities of the differential equations of motion result in losses of accuracy and in considerable increase in computer time when any of the distances between the participating bodies decreases below a certain value. This value is larger than the distance when tidal effects become important, consequently, numerical problems are encountered before the physical picture is changed. Elimination of these singularities by transformations is known as the process of regularization. This paper discusses such transformations and describes in considerable detail the numerical approaches to more accurate and faster integration. The basic ideas of smoothing and regularization are explained and applications are given.

A numerical investigation of the rolling-up of vortex sheets

1980 ◽  
Vol 373 (1752) ◽  
pp. 67-91 ◽  
Keyword(s):  
Numerical Integration ◽  
Numerical Investigation ◽  
Equations Of Motion ◽  
Additional Term ◽  
Vortex Sheet ◽  
Time Step ◽  
Vortex Sheets ◽  
Numerical Instabilities ◽  
Sinusoidal Disturbance ◽  
Induced Velocity

In this paper the development of a vortex sheet due to an initially sinusoidal disturbance is calculated. When determining the induced velocity in points of the vortex sheet, it can be represented by concentrated vortices but it is shown that it is analytically more correct to add an additional term that represents the effect of the immediate neighbourhood of the point considered. The equations of motion were integrated by a Runge-Kutta technique to exclude numerical instabilities. The time step was determined by the requirement that a quantity (Hamiltonian) that remains invariant as a result of the equations of motion, should not change more than a certain amount in the numerical integration of the equations of motion. One difficulty is that if a greater number of concentrated vortices are introduced to represent the vortex sheet, the effect of round-off errors becomes more important. The number of figures retained in the computations limits the number of concentrated vortices. Where the round-off errors have been kept sufficiently small, a process of rolling-up of vorticity clearly occurs. There is no point in pursuing the calculations much beyond this point, first because the representation of the vortex sheet by concentrated vortices becomes more and more inaccurate and secondly because viscosity will have the effect of transforming the rolled-up vortex sheet into a region of vorticity.

scholarly journals Notes regarding the Modeling of the Angle of Attack

2019 ◽  
Vol 304 ◽  
pp. 07012
Author(s):  
Laurentiu Moraru
Keyword(s):  
Numerical Integration ◽  
Equations Of Motion ◽  
Angle Of Attack ◽  
Numerical Algorithms ◽  
Simulated Data ◽  
Flight Simulation ◽  
Approximate Analytical Solutions ◽  
Long Time ◽  
Integration Algorithms ◽  
Nonlinear Equations Of Motion

Numerical integration has become routine for many decades and so has become the numerical integration of the aircraft’s equation of motion. Many numerical algorithms have been used in flight dynamics and the applications of the basic numerical methods to flight simulation have been included in textbooks for a long time. However, many design and/or optimization algorithms rely on analyzing large amounts of simulated data, so analytical algorithms that can provide expedite estimations of the fast varying parameters have been revaluated. The current paper discusses approximate analytical solutions for the angle of attack. Two types of such solutions are discussed. The first model considered originates in the classically linearized equations of motion. The second model discussed was obtained by simplifying the nonlinear equations of motion. The two models are compared against numerical results, provided by classical numerical integration algorithms.

Analysis of an Accelerating Rotor-Bearing System With Flexible Damped Supports

1998 ◽  
Author(s):  
Euro L. Casanova ◽  
Luis U. Medina
Keyword(s):  
Steady State ◽  
Numerical Integration ◽  
Equations Of Motion ◽  
Peak Amplitude ◽  
Graphical Form ◽  
Jeffcott Rotor ◽  
Rotor Bearing ◽  
Flexible Supports ◽  
Steady State Predictions ◽  
Bearing System

This paper deals with the dynamics of an accelerating unbalanced Jeffcott rotor-bearing system mounted on damped, flexible supports. The general equations of motion for such a system are presented and discussed. The rotor response was predicted, via numerical integration, for various cases in runup and rundown conditions and presented in graphical form. The effects of acceleration on the rotor peak amplitude and the speed at which the peak occurs is discussed and compared to steady state predictions.

Formulation of Equations of Motion for Systems Subject to Constraints

1985 ◽  
Vol 52 (2) ◽  
pp. 465-470 ◽  
Author(s):  
C. Wampler ◽  
K. Buffinton ◽  
J. Shu-hui
Keyword(s):  
Numerical Integration ◽  
Explicit Form ◽  
Equations Of Motion ◽  
Constrained Systems ◽  
Dynamical Equations ◽  
Special Cases ◽  
Additional Constraints

A method for constructing equations of motion governing constrained systems is presented. The method, which is particularly useful when equations of motion have already been formulated, and new equations of motion, reflecting the presence of additional constraints are needed, allow the new equations to be written as a recombination of terms comprising the original equations. An explicit form in which the new dynamical equations may be cast for the purpose of numerical integration is developed, along with special cases that demonstrate how the procedure may be simplified in two commonly occurring situations. An illustrative example from the field of robotics is presented, and several areas of application are identified.

scholarly journals The Determination of the Velocities after Impact for the Constrained Bar Problem

2009 ◽  
Vol 2009 ◽  
pp. 1-16
Author(s):  
André Fenili ◽  
Luiz Carlos Gadelha de Souza ◽  
Bernd Schäfer
Keyword(s):  
Mathematical Model ◽  
Numerical Integration ◽  
Equations Of Motion ◽  
Robotic Manipulators ◽  
Robotic Manipulator ◽  
Simple Mathematical Model ◽  
Governing Equations ◽  
Complex Problems ◽  
Constrained Equation

A simple mathematical model for a constrained robotic manipulator is investigated. Besides the fact that this model is relatively simple, all the features present in more complex problems are similar to the ones analyzed here. The fully plastic impact is considered in this paper. Expressions for the velocities of the colliding bodies after impact are developed. These expressions are important in the numerical integration of the governing equations of motion when one must exchange the set of unconstrained equations for the set of constrained equation. The theory presented in this work can be applied to problems in which robots have to follow some prescribed patterns or trajectories when in contact with the environment. It can also de applied to problems in which robotic manipulators must handle payloads.

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