Machine Tool Chatter and Surface Location Error in Milling Processes

2006 ◽  
Vol 128 (4) ◽  
pp. 913-920 ◽  
Author(s):  
Tamás Insperger ◽  
Janez Gradišek ◽  
Martin Kalveram ◽  
Gábor Stépán ◽  
Klaus Winert ◽  
...  
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
High Speed ◽  
Dominant Frequency ◽  
Free Motion ◽  
Solution Technique ◽  
Location Error ◽  
Surface Location Error ◽  
Surface Location ◽  
Delay Differential

A two degree of freedom model of the milling process is investigated. The governing equation of motion is decomposed into two parts: an ordinary differential equation describing the periodic chatter-free motion of the tool and a delay-differential equation describing chatter. The stability chart is derived by using the semi-discretization method for the delay-differential equation corresponding to the chatter motion. The periodic chatter-free motion of the tool and the associated surface location error (SLE) are obtained by a conventional solution technique of ordinary differential equations. It is shown that the SLE is large at the spindle speeds where the ratio of the dominant frequency of the tool and the tooth passing frequency is an integer. This phenomenon is explained by the large amplitude of the periodic chatter-free motion of the tool at these resonant spindle speeds. It is shown that large stable depths of cut with a small SLE can still be attained close to the resonant spindle speeds by using the SLE diagrams associated with stability charts. The results are confirmed experimentally on a high-speed milling center.

Machine Tool Chatter and Surface Quality in Milling Processes

2004 ◽  
Author(s):  
Tama´s Insperger ◽  
Janez Gradisek ◽  
Martin Kalveram ◽  
Ga´bor Ste´pa´n ◽  
Klaus Weinert ◽  
...  
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Periodic Motion ◽  
Milling Process ◽  
Location Error ◽  
Surface Location Error ◽  
Stability Chart ◽  
Stable Periodic ◽  
Surface Location ◽  
Delay Differential

Two degree of freedom model of milling process is investigated. The governing equation of motion is decomposed into two parts: an ordinary differential equation describing the stable periodic motion of the tool and a delay-differential equation describing chatter. Stability chart is derived by using semi-discretization method for the delay-differential equation corresponding to the chatter motion. The stable periodic motion of the tool and the associated surface location error are obtained by a conventional solution technique of ordinary differential equations. Stability chart and surface location error are determined for milling process. It is shown that at spindle speeds, where high depths of cut are available through stable machining, the surface location error is large. The phase portrait of the tool is also analyzed for different spindle speeds. Theoretical predictions are qualitatively confirmed by experiments.

A Numerical Study of Uncertainty in Stability and Surface Location Error in High-Speed Milling

2005 ◽  
Author(s):  
Mohammad H. Kurdi ◽  
Tony L. Schmitz ◽  
Raphael T. Haftka ◽  
Brian P. Mann
Keyword(s):  
High Speed ◽  
Numerical Study ◽  
Removal Rate ◽  
Operating Conditions ◽  
Location Error ◽  
High Speed Milling ◽  
Surface Location Error ◽  
Input Parameters ◽  
Surface Location ◽  
Axial Depth

High-speed milling offers an efficient tool for developing cost effective manufacturing processes with acceptable dimensional accuracy. Realization of these benefits depends on an appropriate selection of preferred operating conditions. In a previous study, optimization was used to find these conditions for two objectives: material removal rate (MRR) and surface location error (SLE), with a Pareto front or tradeoff curve found for the two competing objectives. However, confidence in the optimization results depends on the uncertainty in the input parameters to the milling model (time finite element analysis was applied here for simultaneous prediction of stability and surface location error). In this paper the uncertainty of these input parameters such as cutting force coefficients, tool modal parameters, and cutting parameters is evaluated. The sensitivity of the maximum stable axial depth, blim, to each input parameter at each spindle speed is determined. This enables identification of parameters with high contribution to stability lobe uncertainty. Two methods are used to calculate uncertainty: 1) Monte Carlo simulation; and 2) numerical derivatives of the system eigenvalues. Once the uncertainty in axial depth is calculated, its effect is observed in the MRR and SLE uncertainties. This allows robust optimization that takes into consideration both performance and uncertainty.

Simultaneous Optimization of Removal Rate and Part Accuracy in High-Speed Milling

2004 ◽  
Author(s):  
Mohammad H. Kurdi ◽  
Tony L. Schmitz ◽  
Raphael T. Haftka ◽  
Brian P. Mann
Keyword(s):  
Optimization Algorithm ◽  
High Speed ◽  
Removal Rate ◽  
Spindle Speed ◽  
Successful Implementation ◽  
Location Error ◽  
High Speed Milling ◽  
Surface Location Error ◽  
Surface Location ◽  
Selection Of

High-speed milling provides an efficient method for accurate discrete part fabrication. However, successful implementation requires the selection of appropriate operating parameters. Balancing the multiple process requirements, including high material removal rate, maximum part accuracy, sufficient tool life, chatter avoidance, and adequate surface finish, to arrive at an optimum solution is difficult without the aid of an optimization framework. In this paper an initial effort is made to apply analytical tools to the selection of optimum cutting parameters (spindle speed and depth of cut are considered at this stage). Two objectives are addressed simultaneously, maximum removal rate and minimum surface location error. The Time Finite Element Analysis method is used in the optimization algorithm. Sensitivity of the surface location error to small changes in spindle speed near tooth passing frequencies that are integer fractions of the system’s natural frequency corresponding to the most flexible mode is calculated. Results of the optimization algorithm are verified by experiment.

scholarly journals Smooth Stable Manifold for Delay Differential Equations with Arbitrary Growth Rate

Axioms ◽  
2021 ◽  
Vol 10 (2) ◽  
pp. 105
Author(s):  
Lokesh Singh ◽  
Dhirendra Bahuguna
Keyword(s):  
Differential Equation ◽  
Growth Rate ◽  
Invariant Manifold ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Stable Manifold ◽  
Exponential Dichotomy ◽  
Delay Differential ◽  
Stable Invariant ◽  
Nonuniform Exponential Dichotomy

In this article, we construct a C1 stable invariant manifold for the delay differential equation x′=Ax(t)+Lxt+f(t,xt) assuming the ρ-nonuniform exponential dichotomy for the corresponding solution operator. We also assume that the C1 perturbation, f(t,xt), and its derivative are sufficiently small and satisfy smoothness conditions. To obtain the invariant manifold, we follow the method developed by Lyapunov and Perron. We also show the dependence of invariant manifold on the perturbation f(t,xt).

scholarly journals Oscillatory Behaviour of a First-Order Neutral Differential Equation in relation to an Old Open Problem

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
K. C. Panda ◽  
R. N. Rath ◽  
S. K. Rath
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Oscillatory Behaviour ◽  
Neutral Delay ◽  
First Order ◽  
Neutral Delay Differential Equation ◽  
Delay Differential ◽  
Oscillation And Nonoscillation

In this paper, we obtain sufficient conditions for oscillation and nonoscillation of the solutions of the neutral delay differential equation yt−∑j=1kpjtyrjt′+qtGygt−utHyht=ft, where pj and rj for each j and q,u,G,H,g,h, and f are all continuous functions and q≥0,u≥0,ht<t,gt<t, and rjt<t for each j. Further, each rjt, gt, and ht⟶∞ as t⟶∞. This paper improves and generalizes some known results.

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