A varying time-step explicit numerical integration algorithm for solving motion equation

2005 ◽  
Vol 18 (2) ◽  
pp. 239-244
Author(s):  
Zheng-hua Zhou ◽  
Yu-huan Wang ◽  
Quan Liu ◽  
Xiao-tao Yin ◽  
Cheng Yang
Keyword(s):  
Numerical Integration ◽  
Motion Equation ◽  
Time Step ◽  
Integration Algorithm

Numerical integration algorithm of a new model for metal plasticity and fracture including pressure and lode angle dependence

2009 ◽  
Vol 2 (S1) ◽  
pp. 443-446 ◽  
Author(s):  
L. Malcher ◽  
F. M. Andrade Pires ◽  
J. M. A. César de Sá ◽  
F. X. C. Andrade
Keyword(s):  
Numerical Integration ◽  
Integration Algorithm ◽  
New Model ◽  
Metal Plasticity ◽  
Angle Dependence ◽  
Lode Angle

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.

Parametric studies on dynamic stiffness of ball bearings supporting a flexible rotor

2019 ◽  
Vol 25 (15) ◽  
pp. 2175-2188 ◽  
Author(s):  
Tikam Chand Gupta
Keyword(s):  
Numerical Integration ◽  
Dynamic Stiffness ◽  
Mathematical Formulation ◽  
Rolling Element Bearing ◽  
Radial Clearance ◽  
Integration Scheme ◽  
Time Step ◽  
Average Value ◽  
Bearing Stiffness ◽  
Time Invariant

Most of the researchers in the field of dynamics of the rolling element bearing have considered bearing stiffness as time invariant and/or not related to dynamics of the bearing. In the present paper, the bearing stiffness has been taken as function of dynamic response at every time step of numerical simulation and a detailed parametric study is performed to investigate the effect of flexibility of the rotor shaft, rotational speed, and internal radial clearance on the instantaneous and average value of dynamic stiffness of the ball bearing. The mathematical formulation is based on the Timoshenko beam finite element theory. Gravity and bearing forces are considered as external forces acting on a free-free flexible shaft. A stable Newmark- β numerical integration scheme coupled with Newton–Raphson method is used for numerical integration and for convergence to an accurate value of bearing stiffness. The results showing the variation of different components of bearing stiffness as a function of time-invariant parameters has improved the understanding of the dynamic behavior of the bearing during motion. The variation pattern of bearing stiffness coefficients is observed to be sensitive to direction of rotation. The amplitude of periodic change of these coefficients increases with the increase of the stiffness ratio of shaft and the decrease of radial clearance.

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.

A Precise and Stiffly Stable Time Integration Method for Vibration Equations

2001 ◽  
Author(s):  
Takeshi Fujikawa ◽  
Etsujiro Imanishi
Keyword(s):  
Integration Method ◽  
Time Integration ◽  
Low Frequency ◽  
Quadratic Function ◽  
Time Step ◽  
Integration Algorithm ◽  
Time Integration Method ◽  
Time Integration Algorithm ◽  
Time Integration Methods ◽  
Motion Problems

Abstract A method of time integration algorithm is presented for solving stiff vibration and motion problems. It is absolutely stable, numerically dissipative, and much accurate than other dissipative time integration methods. It achieves high-frequency dissipation, while minimizing unwanted low-frequency dissipation. In this method change of acceleration during time step is expressed as quadratic function including some parameters, whose appropriate values are determined through numerical investigation. Two calculation examples are demonstrated to show the usefulness of this method.

Stability of an explicit multi-time step integration algorithm for linear structural dynamics equations

1996 ◽  
Vol 18 (3) ◽  
pp. 236-244 ◽  
Author(s):  
P. Smolinski ◽  
S. Sleith ◽  
T. Belytschko
Keyword(s):  
Structural Dynamics ◽  
Time Step ◽  
Integration Algorithm

scholarly journals Approximate Solutions of Delay Differential Equations with Constant and Variable Coefficients by the Enhanced Multistage Homotopy Perturbation Method

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
D. Olvera ◽  
A. Elías-Zúñiga ◽  
L. N. López de Lacalle ◽  
C. A. Rodríguez
Keyword(s):  
Differential Equations ◽  
Perturbation Method ◽  
Numerical Integration ◽  
Delay Differential Equations ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Variable Coefficients ◽  
Integration Algorithm ◽  
Homotopy Perturbation ◽  
Delay Differential

We expand the application of the enhanced multistage homotopy perturbation method (EMHPM) to solve delay differential equations (DDEs) with constant and variable coefficients. This EMHPM is based on a sequence of subintervals that provide approximate solutions that require less CPU time than those computed from the dde23 MATLAB numerical integration algorithm solutions. To address the accuracy of our proposed approach, we examine the solutions of several DDEs having constant and variable coefficients, finding predictions with a good match relative to the corresponding numerical integration solutions.

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