Numerical Integration Method to Determine Periodic Solutions of Nonlinear Systems

Volume 1: 20th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C ◽

10.1115/detc2005-84529 ◽

2005 ◽

Author(s):

Takao Torii ◽

Nobuyoshi Morita ◽

Kimihiko Yasuda

Keyword(s):

Nonlinear Systems ◽

Numerical Integration ◽

Integration Method ◽

Piecewise Linear ◽

Special Treatment ◽

Numerical Integration Method ◽

Impact Damper ◽

Coulomb’S Friction ◽

Linear Spring ◽

Spring System

A new method to obtain periodic solution of nonlinear systems is proposed by combining secant method with numerical integration. This method can be applied to nonlinear systems with discontinuous characteristics without special treatment. To check the validity of the proposed method, we applied this method to nonlinear systems with discontinuous characteristics, such as piecewise linear spring system, system with Coulomb’s friction and impact damper system. Comparing the results by the proposed method with the results by simple numerical integration or experimental results, the validity of the proposed method is confirmed.

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Numerical Integration Method for Stability Analysis of Milling With Variable Spindle Speeds

Journal of Vibration and Acoustics ◽

10.1115/1.4031617 ◽

2015 ◽

Vol 138 (1) ◽

Cited By ~ 18

Author(s):

Ye Ding ◽

Jinbo Niu ◽

LiMin Zhu ◽

Han Ding

Keyword(s):

Differential Equation ◽

Stability Analysis ◽

Numerical Integration ◽

Integration Method ◽

Transcendental Equation ◽

Spindle Speed ◽

Machining Parameters ◽

Numerical Integration Method ◽

Stability Diagrams ◽

Time Period

A semi-analytical method is presented in this paper for stability analysis of milling with a variable spindle speed (VSS), periodically modulated around a nominal spindle speed. Taking the regenerative effect into account, the dynamics of the VSS milling is governed by a delay-differential equation (DDE) with time-periodic coefficients and a time-varying delay. By reformulating the original DDE in an integral-equation form, one time period is divided into a series of subintervals. With the aid of numerical integrations, the transition matrix over one time period is then obtained to determine the milling stability by using Floquet theory. On this basis, the stability lobes consisting of critical machining parameters can be calculated. Unlike the constant spindle speed (CSS) milling, the time delay for the VSS is determined by an integral transcendental equation which is accurately calculated with an ordinary differential equation (ODE) based method instead of the formerly adopted approximation expressions. The proposed numerical integration method is verified with high computational efficiency and accuracy by comparing with other methods via a two-degree-of-freedom milling example. With the proposed method, this paper details the influence of modulation parameters on stability diagrams for the VSS milling.

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The Numerical Methods of Peridynamics Theory Used in Failure Analysis of Materials

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.1030-1032.223 ◽

2014 ◽

Vol 1030-1032 ◽

pp. 223-227

Author(s):

Lin Fan ◽

Song Rong Qian ◽

Teng Fei Ma

Keyword(s):

Differential Equation ◽

Integral Equation ◽

Numerical Integration ◽

Integration Method ◽

Equation Of Motion ◽

Numerical Integration Method ◽

Volume Integral ◽

Volume Integral Equation ◽

Method Of Molecular Dynamics ◽

Classic Theory

In order to analysis the force situation of the material which is discontinuity,we can used the new theory called peridynamics to slove it.Peridynamics theory is a new method of molecular dynamics that develops very quickly.Peridynamics theory used the volume integral equation to constructed the model,used the volume integral equation to calculated the PD force in the horizon.So It doesn’t need to assumed the material’s continuity which must assumed that use partial differential equation to formulates the equation of motion. Destruction and the expend of crack which have been included in the peridynamics’ equation of motion.Do not need other additional conditions.In this paper,we introduce the peridynamics theory modeling method and introduce the relations between peridynamics and classic theory of mechanics.We also introduce the numerical integration method of peridynamics.Finally implementation the numerical integration in prototype microelastic brittle material.Through these work to show the advantage of peridynamics to analysis the force situation of the material.

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