scholarly journals On the bistable zone of milling processes

2015 ◽  
Vol 373 (2051) ◽  
pp. 20140409 ◽  
Author(s):  
Zoltan Dombovari ◽  
Gabor Stepan
Keyword(s):  
Periodic Solution ◽  
Domain Of Attraction ◽  
Cutting Parameters ◽  
Modal Testing ◽  
Structural Behaviour ◽  
Infinite Dimensional ◽  
Dimensional Phase Space ◽  
Periodic Delay ◽  
Delay Differential ◽  
Time Periodic

A modal-based model of milling machine tools subjected to time-periodic nonlinear cutting forces is introduced. The model describes the phenomenon of bistability for certain cutting parameters. In engineering, these parameter domains are referred to as unsafe zones, where steady-state milling may switch to chatter for certain perturbations. In mathematical terms, these are the parameter domains where the periodic solution of the corresponding nonlinear, time-periodic delay differential equation is linearly stable, but its domain of attraction is limited due to the existence of an unstable quasi-periodic solution emerging from a secondary Hopf bifurcation. A semi-numerical method is presented to identify the borders of these bistable zones by tracking the motion of the milling tool edges as they might leave the surface of the workpiece during the cutting operation. This requires the tracking of unstable quasi-periodic solutions and the checking of their grazing to a time-periodic switching surface in the infinite-dimensional phase space. As the parameters of the linear structural behaviour of the tool/machine tool system can be obtained by means of standard modal testing, the developed numerical algorithm provides efficient support for the design of milling processes with quick estimates of those parameter domains where chatter can still appear in spite of setting the parameters into linearly stable domains.

scholarly journals A Method for Stability Analysis of Periodic Delay Differential Equations with Multiple Time-Periodic Delays

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Gang Jin ◽  
Houjun Qi ◽  
Zhanjie Li ◽  
Jianxin Han ◽  
Hua Li
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Spindle Speed ◽  
Multiple Time ◽  
Variable Pitch ◽  
Periodic Delay ◽  
The Stability ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

Delay differential equations (DDEs) are widely utilized as the mathematical models in engineering fields. In this paper, a method is proposed to analyze the stability characteristics of periodic DDEs with multiple time-periodic delays. Stability charts are produced for two typical examples of time-periodic DDEs about milling chatter, including the variable-spindle speed milling system with one-time-periodic delay and variable pitch cutter milling system with multiple delays. The simulations show that the results gained by the proposed method are in close agreement with those existing in the past literature. This indicates the effectiveness of our method in terms of time-periodic DDEs with multiple time-periodic delays. Moreover, for milling processes, the proposed method further provides a generalized algorithm, which possesses a good capability to predict the stability lobes for milling operations with variable pitch cutter or variable-spindle speed.

Computation and Stability Analysis of Solutions of Periodic Delay Differential Algebraic Equations

2005 ◽  
Author(s):  
Koen Verheyden ◽  
Kurt Lust ◽  
Dirk Roose
Keyword(s):  
Stability Analysis ◽  
Numerical Methods ◽  
Periodic Solutions ◽  
Numerical Algorithms ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Periodic Delay ◽  
Delay Differential ◽  
Time Periodic ◽  
Periodic Delays

This paper is concerned with the numerical computation, continuation and stability analysis of periodic solutions of periodic delay differential algebraic equations. We consider systems with a time-periodic right hand side function and time-periodic delays. We introduce numerical algorithms based on collocation to compute periodic solutions and their stability. The presented methods combine knowledge from numerical methods for delay equations and differential algebraic equations. Our algorithms are illustrated with numerical results for two models.

Nonlinear Regenerative Machine Tool Vibrations

1997 ◽  
Author(s):  
Gábor Stépán ◽  
Tamas Kalmar-Nagy
Keyword(s):  
Machine Tool ◽  
Analytical Formula ◽  
Chip Thickness ◽  
Domain Of Attraction ◽  
Equation Model ◽  
Center Manifold Reduction ◽  
Nonlinear Part ◽  
Dimensional Phase Space ◽  
Lengthy Calculation ◽  
Delay Differential

Abstract The existence and the nature of the Hopf bifurcation is presented in the delay-differential equation model of the so-called regenerative machine tool vibration. The relevant nonlinearity is considered at the cutting force dependence on the chip thickness. The delayed terms show a special algebraic structure in the nonlinear part of the equation of motion. This results in a surprisingly simple and useful analytical formula in the end of the lengthy calculation based on center manifold reduction in the corresponding infinite dimensional phase space. The result gives a simple way to estimate the domain of attraction of the stable stationary cutting as well as an estimation of that technological parameter domain, where the cutting is globally stable.

Period Doubling Bifurcation and Center Manifold Reduction in a Time-periodic and Time-delayed Model of Machining

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1169-1187 ◽  
Author(s):  
Róbert Szalai ◽  
Gábor Stépán
Keyword(s):  
Center Manifold ◽  
Period Doubling ◽  
Characteristic Parameter ◽  
Period Doubling Bifurcation ◽  
Numerical Continuation ◽  
Machining Processes ◽  
Center Manifold Reduction ◽  
Periodic Delay ◽  
Delay Differential ◽  
Time Periodic

A closed-form calculation is presented for the analysis of the period-doubling bifurcation in the time-periodic delay-differential equation model of interrupted machining processes such as milling where the nonlinearity is essentially nonsymmetric. We prove the subcritical sense of this period-doubling bifurcation and approximate the emerging period-two oscillations by the Lyapunov—Perron method for computing the center manifold and by calculating the Poincaré—Lyapunov constant of the bifurcation analytically at certain characteristic parameter values. The existence of the unstable period-two oscillations around the stable stationary cutting is confirmed using a numerical continuation algorithm developed for time-periodic delay-differential equations.

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