Stability Analysis Method for Periodic Delay Differential Equations with Multiple Distributed and Time-Varying Delays

Dynamic stability problems leading to delay differential equations (DDEs) are found in many different fields of science and engineering. In this paper, a method for stability analysis of periodic DDEs with multiple distributed and time-varying delays is proposed, based on the well-known semidiscretization method. In order to verify the correctness of the proposed method, two typical application examples, i.e., milling process with a variable helix cutter and milling process with variable spindle speed, which can be, respectively, described by DDEs with the multidistributed and time-varying delays are considered. Then, comparisons with prior methods for stability prediction are made to verify the accuracy and efficiency of the proposed approach. As far as the milling process is concerned, the proposed method supplies a generalized algorithm to analyze the stability of the single milling systems associated with variable pith cutter, variable helix cutter, or variable spindle speed; it also can be utilized to analyze the combined systems of the aforementioned cases.

[1]  C. Hsu,et al.  Stability Criteria for Second-Order Dynamical Systems With Time Lag , 1966 .

[2]  Qichang Zhang,et al.  Stability prediction of milling process with variable pitch and variable helix cutters , 2014 .

[3]  Neil D. Sims,et al.  Analytical prediction of chatter stability for variable pitch and variable helix milling tools , 2008 .

[4]  Gábor Stépán,et al.  Semi‐discretization method for delayed systems , 2002 .

[5]  Qichang Zhang,et al.  Stability prediction for milling process with multiple delays using an improved semi‐discretization method , 2016 .

[6]  Haitao Ma,et al.  Stability of linear time‐periodic delay‐differential equations via Chebyshev polynomials , 2004 .

[7]  Gilles Dessein,et al.  On the stability of high-speed milling with spindle speed variation , 2010 .

[8]  Jianxin Han,et al.  Dynamic modeling and stability analysis for the combined milling system with variable pitch cutter and spindle speed variation , 2018, Commun. Nonlinear Sci. Numer. Simul..

[10]  P. Niamsup,et al.  Delay-Dependent Robust H∞ Performance for Uncertain Neutral Systems with Mixed Time-Varying Delays and Nonlinear Perturbations , 2018, Mathematical Problems in Engineering.

[12]  Berend Denkena,et al.  Stable islands in the stability chart of milling processes due to unequal tooth pitch , 2011 .

[13]  Keith Ridgway,et al.  Modelling of the stability of variable helix end mills , 2007 .

[14]  Tamás Insperger,et al.  Full-discretization and semi-discretization for milling stability prediction: Some comments , 2010 .

[15]  Gábor Stépán,et al.  State Dependent Regenerative Delay in Milling Processes , 2005 .

[16]  Nejat Olgac,et al.  Dynamics and Stability of Variable-pitch Milling , 2007 .

[17]  Gang Jin,et al.  A frequency-domain solution for efficient stability prediction of variable helix cutters milling , 2014 .

[18]  Ye Ding,et al.  Variable-step integration method for milling chatter stability prediction with multiple delays , 2011 .

[19]  Gábor Stépán,et al.  Updated semi‐discretization method for periodic delay‐differential equations with discrete delay , 2004 .

[20]  Yusuf Altintas,et al.  Analytical Stability Prediction and Design of Variable Pitch Cutters , 1998, Manufacturing Science and Engineering.

[21]  Zoltan Dombovari,et al.  The Effect of Helix Angle Variation on Milling Stability , 2012 .

[22]  Han Ding,et al.  A full-discretization method for prediction of milling stability , 2010 .

[23]  Non-differentiable Delay-interval-dependent Exponentially Passive Conditions for Certain Neutral Integro-differential Equations with Time-varying Delays , 2020 .

[24]  Neil D. Sims,et al.  Optimisation of variable helix tool geometry for regenerative chatter mitigation , 2011 .

[25]  Zoltan Dombovari,et al.  The effect of serration on mechanics and stability of milling cutters , 2010 .

[26]  Min Wan,et al.  A unified stability prediction method for milling process with multiple delays , 2010 .

[27]  B. Mann,et al.  Stability of Interrupted Cutting by Temporal Finite Element Analysis , 2003 .

[28]  A. Galip Ulsoy,et al.  Delay differential equations via the matrix Lambert W function and bifurcation analysis: application to machine tool chatter. , 2007, Mathematical biosciences and engineering : MBE.

[29]  Gábor Stépán,et al.  Semi-Discretization for Time-Delay Systems: Stability and Engineering Applications , 2011 .

[30]  E. Dowell,et al.  The High Dimensional Harmonic Balance Analysis of a Second-Order Delay-Differential Equations , 2007 .

[31]  Erhan Budak,et al.  An analytical design method for milling cutters with nonconstant pitch to increase stability, Part I: Theory , 2003 .

[32]  Firas A. Khasawneh,et al.  Stability of delay integro-differential equations using a spectral element method , 2011, Math. Comput. Model..

[33]  Pankaj Wahi,et al.  Galerkin Projections for Delay Differential Equations , 2005 .

[34]  K. Mukdasai,et al.  A Novel Delay-Dependent Asymptotic Stability Conditions for Differential and Riemann-Liouville Fractional Differential Neutral Systems with Constant Delays and Nonlinear Perturbation , 2020, Mathematics.

[35]  Gábor Stépán,et al.  Extension of the spectral element method for stability analysis of time-periodic delay-differential equations with multiple and distributed delays , 2016, Commun. Nonlinear Sci. Numer. Simul..

[36]  Nejat Olgac,et al.  “Delay Scheduling”: A New Concept for Stabilization in Multiple Delay Systems , 2005 .