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

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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Stability Analysis of Milling Processes With Periodic Spindle Speed Variation Via the Variable-Step Numerical Integration Method

Journal of Manufacturing Science and Engineering ◽

10.1115/1.4033043 ◽

2016 ◽

Vol 138 (11) ◽

Cited By ~ 13

Author(s):

Jinbo Niu ◽

Ye Ding ◽

LiMin Zhu ◽

Han Ding

Keyword(s):

Stability Analysis ◽

Numerical Integration ◽

Integration Method ◽

Spindle Speed ◽

Numerical Integration Method ◽

Speed Variation ◽

Spindle Speed Variation ◽

Variable Step ◽

Modulation Schemes ◽

The Stability

This paper proposes a general method for the stability analysis and parameter optimization of milling processes with periodic spindle speed variation (SSV). With the aid of Fourier series, the time-variant spindle speeds of different periodic modulation schemes are unified into one framework. Then the time-varying delay is derived implicitly and calculated efficiently using an accurate ordinary differential equation (ODE) based algorithm. After incorporating the unified spindle speed and time delay into the dynamic model, a Floquet theory based variable-step numerical integration method (VNIM) is presented for the stability analysis of variable spindle speed milling processes. By comparison with other methods, such as the semi-discretization method and the constant-step numerical integration method, the proposed method has the advantages of high computational accuracy and efficiency. Finally, different spindle speed modulation schemes are compared and the modulation parameters are optimized with the aid of three-dimensional stability charts obtained using the proposed VNIM.

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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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