Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation

Deshmukh, Venkatesh; Butcher, Eric A.; Bueler, Ed

doi:10.1007/s11071-007-9266-6

Cited by 27 publications

(13 citation statements)

References 28 publications

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“…The proposed method easily extends to systems with many degrees of freedom, and it produces stability charts with high speed and accuracy for a given parameter range. Other than a preliminary version of this work in [39], this algorithm has also been recently used for dimensional reduction of nonlinear periodic DDEs [40], and for parameter estimation in nonlinear time-varying ODEs [41].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels

Butcher

Bobrenkov

Bueler

et al. 2009

Journal of Computational and Nonlinear Dynamics

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112

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In this paper the dynamic stability of the milling process is investigated through a single degree-of-freedom model by determining the regions where chatter (unstable) vibrations occur in the two-parameter space of spindle speed and depth of cut. Dynamic systems like milling are modeled by delay-differential equations (DDEs) with timeperiodic coefficients. A new approximation technique for studying the stability properties of such systems is presented. The approach is based on the properties of Chebyshev polynomials and a collocation expansion of the solution. The collocation points are the extreme points of a Chebyshev polynomial of high degree. Specific cutting force profiles and stability charts are presented for the up-and down-milling cases of one or two cutting teeth and various immersion levels with linear and nonlinear regenerative cutting forces.The unstable regions due to both secondary Hopf and flip (period-doubling) bifurcations CND-08-1016, Butcher 2 are found, and an in-depth investigation of the optimal stable immersion levels for downmilling in the vicinity of where the average cutting force changes sign is presented.

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Section: Introductionmentioning

confidence: 99%

Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels

Butcher

Bobrenkov

Bueler

et al. 2009

Journal of Computational and Nonlinear Dynamics

Self Cite

112

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“…A study on stability and performance of feedback controls with multiple time delays is reported in [6] by considering the roots of the closed-loop characteristic equation. A method using Chebyshev polynomials to approximate general nonlinear functions of time has been developed to handle linear and nonlinear time-delayed dynamical systems with periodic coefficients [7][8][9][10]. The method has also been applied to study optimal control problems.…”

Section: Introductionmentioning

confidence: 99%

Control studies of time-delayed dynamical systems with the method of continuous time approximation

Sun

Song

2009

Communications in Nonlinear Science and Numerical Simulation

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“…The solution method illustrated here is the perturbation series method (Nayfeh, 1981), which is modified from Deshmukh et al (2006). It applies to systems of the form of equation 29.…”

Section: Perturbation Series Solutionmentioning

confidence: 99%

Approximate Stability Analysis and Computation of Solutions of Nonlinear Delay Differential Algebraic Equations with Time Periodic Coefficients

Deshmukh

2010

Journal of Vibration and Control

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Approximate stability analysis of nonlinear delay differential algebraic equations (DDAEs) with periodic coefficients is proposed with a geometric interpretation of evolution of the linearized system. Firstly, a numerical algorithm based on direct integration by expansion in terms of Chebyshev polynomials is derived for linear analysis. The proposed algorithm is shown to have deeper connections with and be computationally less cumbersome than the solution of the underlying semi-explicit system via a similarity transformation. The stability of time periodic DDAE systems is characterized by the spectral radius of a “monodromy matrix”, which is a finite-dimensional approximation of a compact infinite-dimensional operator. The monodromy matrix is essentially a map of the Chebyshev coefficients (or collocation vector) of the state from the delay interval to the next adjacent interval of time. The computations are entirely performed with the original system to avoid cumbersome transformations associated with the semi-explicit form of the system. Next, two computational algorithms, the first based on perturbation series and the second based on Chebyshev spectral collocation, are detailed to obtain solutions of nonlinear DDAEs with periodic coefficients for consistent initial functions.

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Dimensional reduction of nonlinear delay differential equations with periodic coefficients using Chebyshev spectral collocation

Cited by 27 publications

References 28 publications

Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels

Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels

Control studies of time-delayed dynamical systems with the method of continuous time approximation

Approximate Stability Analysis and Computation of Solutions of Nonlinear Delay Differential Algebraic Equations with Time Periodic Coefficients

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