On Passive Non-Iterative Variable-Step Numerical Integration of Mechanical Systems for Haptic Rendering

2008 ◽  
Author(s):  
Dongjun Lee ◽  
Ke Huang
Keyword(s):  
Numerical Integration ◽  
Mechanical Systems ◽  
Haptic Rendering ◽  
Haptic Device ◽  
Integration Step ◽  
Constant Mass ◽  
Supply Rate ◽  
Integration Algorithm ◽  
Collision Handling ◽  
Variable Step

This paper consists of three parts. First, with a slightly-different, yet, more physically-plausible, discrete supply-rate (i.e. power), we propose a non-iterative (i.e. fast) and variable-step numerical integration algorithm for (scalar) discrete-passive mechanical systems, consisting of constant mass and damper, and a certain class of nonlinear spring. In the second part, we propose a fast passive collision handling algorithm with a spring-damper type virtual wall, which, to detect exact time of contacts, requires at most three intermediate non-iterative computations within each integration-step. We then propose a way of how to passively connect this discrete-passive, non-iterative, and variable-step mechanical integrators (with passive collision handling) to a continuous haptic device.

Dynamic Analysis of Mechanical Systems With Intermittent Motion

1982 ◽  
Vol 104 (4) ◽  
pp. 778-784 ◽  
Author(s):  
R. A. Wehage ◽  
E. J. Haug
Keyword(s):  
Dynamic Analysis ◽  
Numerical Integration ◽  
Large Scale ◽  
Mechanical Systems ◽  
Integration Algorithm ◽  
Large Scale Systems ◽  
Jump Discontinuities ◽  
Impulsive Forces ◽  
Computer Generation ◽  
Intermittent Motion

A method is presented for dynamic analysis of systems with impulsive forces, impact, discontinuous constraints, and discontinuous velocities. A method of computer generation of the equations of planar motion and impulse-momentum relations that define jump discontinuities in system velocity for large scale systems is presented. An event predictor, working in conjunction with a new numerical integration algorithm, efficiently controls the numerical integration and allows for automatic equation reformulation. A weapon mechanism and a trip plow are simulated using the method to illustrate its capabilities.

scholarly journals A Transient Stability Numerical Integration Algorithm for Variable Step Sizes Based on Virtual Input

Energies ◽  
2017 ◽  
Vol 10 (11) ◽  
pp. 1736 ◽  
Author(s):  
Yifan Gao ◽  
Jianquan Wang ◽  
Tannan Xiao ◽  
Daozhuo Jiang
Keyword(s):  
Numerical Integration ◽  
Transient Stability ◽  
Integration Algorithm ◽  
Variable Step

Stabilization Method for Numerical Integration of Multibody Mechanical Systems

1998 ◽  
Vol 120 (4) ◽  
pp. 565-572 ◽  
Author(s):  
Shih-Tin Lin ◽  
Ming-Chong Hong
Keyword(s):  
Numerical Integration ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Constraint Equation ◽  
Differential Algebraic Equations ◽  
Integration Step ◽  
Algebraic Equations ◽  
Stabilization Method ◽  
Step Size ◽  
Integration Methods

The object of this study is to solve the stability problem for the numerical integration of constrained multibody mechanical systems. The dynamic equations of motion of the constrained multibody mechanical system are mixed differential-algebraic equations (DAE). In applying numerical integration methods to this equation, constrained equations and their first and second derivatives must be satisfied simultaneously. That is, the generalized coordinates and their derivatives are dependent. Direct integration methods do not consider this dependency and constraint violation occurs. To solve this problem, Baumgarte proposed a constraint stabilization method in which a position and velocity terms were added in the second derivative of the constraint equation. The disadvantage of this method is that there is no reliable method for selecting the coefficients of the position and velocity terms. Improper selection of these coefficients can lead to erroneous results. In this study, stability analysis methods in digital control theory are used to solve this problem. Correct choice of the coefficients for the Adams method are found for both fixed and variable integration step size.

scholarly journals The Use of Integrals in Numerical Integrations of theN-Body Problem

1971 ◽  
Vol 10 ◽  
pp. 40-51
Author(s):  
Paul E. Nacozy
Keyword(s):  
Angular Momentum ◽  
Numerical Integration ◽  
Degrees Of Freedom ◽  
Body Problem ◽  
Center Of Mass ◽  
Integration Step ◽  
Gravitational System ◽  
Systems Of Differential Equations ◽  
Numerical Integrations ◽  
Original Number

AbstractThe numerical integration of systems of differential equations that possess integrals is often approached by using the integrals to reduce the number of degrees of freedom or by using the integrals as a partial check on the resulting solution, retaining the original number of degrees of freedom.Another use of the integrals is presented here. If the integrals have not been used to reduce the system, the solution of a numerical integration may be constrained to remain on the integral surfaces by a method that applies corrections to the solution at each integration step. The corrections are determined by using linearized forms of the integrals in a least-squares procedure.The results of an application of the method to numerical integrations of a gravitational system of 25-bodies are given. It is shown that by using the method to satisfy exactly the integrals of energy, angular momentum, and center of mass, a solution is obtained that is more accurate while using less time of calculation than if the integrals are not satisfied exactly. The relative accuracy is ascertained by forward and backward integrations of both the corrected and uncorrected solutions and by comparison with more accurate integrations using reduced step-sizes.

Sensitivity Analysis and Optimization of Mechanical System Design

1988 ◽  
Author(s):  
W. J. Langner
Keyword(s):  
Linear Equation ◽  
Mechanical System ◽  
Numerical Integration ◽  
Equation System ◽  
Sensitivity Analyses ◽  
Design Parameters ◽  
Governing Equations ◽  
Integration Algorithm ◽  
Linear Equation System ◽  
The Matrix

Abstract The paper follows studies on simulation of three-dimensional mechanical dynamic systems with the help of sparse matrix and stiff integration numerical algorithms. For sensitivity analyses and the application of numerical optimization procedures it is substantial to calculate the effect of design parameters on the system behaviour by means of derivatives of state variables with respect to the design parameters. For static and quasi static analyses the computation of these derivatives from the governing equations leads to a linear equation system. The matrix of this set of linear equations shows to be the Jacobian matrix required in the numerical integration process solving the system of governing equations for the mechanical system. Thus the factorization of the matrix perfomed by the numerical integration algorithm can be reused solving the linear equation system for the state variable sensitivities. Some example demonstrate the simplicity of building the right hand sides of the linear equation system. Also it is demonstrated that the procedure proposed neatly fits into a modular concept for simulation model building and analysis.

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