scholarly journals Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter

2007 ◽  
Vol 4 (2) ◽  
pp. 355-368 ◽  
Keyword(s):  
Differential Equations ◽  
Machine Tool ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Lambert W Function ◽  
Machine Tool Chatter ◽  
The Matrix ◽  
Analysis Application ◽  
Delay Differential ◽  
Tool Chatter

scholarly journals On the Convergence of the Matrix Lambert W Approach to Solution of Systems of Delay Differential Equations

2018 ◽  
Author(s):  
A. Galip Ulsoy ◽  
Rita Gitik
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Iteration Method ◽  
Delay Differential Equations ◽  
Function Method ◽  
Lambert W Function ◽  
The Matrix ◽  
Iterative Solutions ◽  
Convergence Problems ◽  
Delay Differential

Convergence aspects of the matrix Lambert W function method for solving systems of delay differential equations (DDEs) are considered. Recent research results show that convergence problems can occur with certain DDEs when using the well-established Q-iteration approach. A complementary, and recently proposed, W-iteration approach is shown to converge even on systems where the Q-iteration fails. Furthermore, the role played by the branch numbers k = -∞ .. -2, -1, 0, 1, 2 , .. ∞ of the matrix Lambert W function, Wk, in terms of initializing the iterative solutions, is also discussed and elucidated. Several second order examples, known to have convergence problems with Q-iteration, are readily solved by W-iteration. Examples of third and fourth order DDEs show that the W-iteration method is also effective on higher-order systems.

scholarly journals On the Convergence of the Matrix Lambert W Approach to Solution of Systems of Delay Differential Equations

2019 ◽  
Vol 142 (2) ◽  
Author(s):  
A. Galip Ulsoy ◽  
Rita Gitik
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Delay Differential Equations ◽  
Function Method ◽  
Higher Order ◽  
Lambert W Function ◽  
The Matrix ◽  
Iterative Solutions ◽  
Convergence Problems ◽  
Delay Differential

Abstract Convergence of the matrix Lambert W function method for solving systems of delay differential equations (DDEs) is considered. Recent research shows that convergence problems occur with certain DDEs when using the well-established Q-iteration approach. A complementary, and recently proposed, W-iteration approach is shown to converge even on systems where Q-iteration fails. Furthermore, the role played by the branch numbers k = −∞ … −1, 0, 1, … ∞ of the matrix Lambert W function, Wk, in terms of initializing the iterative solutions, is also discussed and elucidated. Several second-order examples, known to have convergence problems with Q-iteration, are readily solved by W-iteration. Examples of third- and fourth-order DDEs show that W-iteration is also effective on higher-order systems.

Chatter Stability Analysis Using the Matrix Lambert Function and Bifurcation Analysis

2006 ◽  
Author(s):  
Sun Yi ◽  
Patrick W. Nelson ◽  
A. Galip Ulsoy
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Lambert Function ◽  
Chatter Stability ◽  
Approximate Methods ◽  
Linear Delay ◽  
The Matrix ◽  
The Stability ◽  
Delay Differential

We investigate the stability of the regenerative machine tool chatter problem, in a turning process modeled using delay differential equations (DDEs). An approach using the matrix Lambert function for the analytical solution to systems to delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert function, known to be useful for solving scalar first order DDEs, has recently been extended to a matrix Lambert function approach to solve systems of DDEs. The essential advantage of the matrix Lambert approach is not only the similarity to the concept of the state transition matrix in linear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy, and certain other advantages, when compared to traditional graphical, computational and approximate methods.

NUMERICAL BIFURCATION ANALYSIS OF DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY

2001 ◽  
Vol 11 (03) ◽  
pp. 737-753 ◽  
Author(s):  
TATYANA LUZYANINA ◽  
KOEN ENGELBORGHS ◽  
DIRK ROOSE
Keyword(s):  
Differential Equations ◽  
Bifurcation Analysis ◽  
Delay Differential Equations ◽  
Constant Delay ◽  
State Dependent ◽  
State Dependent Delay ◽  
Numerical Bifurcation ◽  
Delay Differential ◽  
Steady State Solutions ◽  
Numerical Bifurcation Analysis

In this paper we apply existing numerical methods for bifurcation analysis of delay differential equations with constant delay to equations with state-dependent delay. In particular, we study the computation, continuation and stability analysis of steady state solutions and periodic solutions. We collect the relevant theory and describe open theoretical problems in the context of bifurcation analysis. We present computational results for two examples and compare with analytical results whenever possible.

Nonlinear Delay Equations With Fluctuating Delay: Application to Regenerative Chatter

2002 ◽  
Author(s):  
Ali Demir ◽  
N. Sri Namachchivaya ◽  
W. F. Langford
Keyword(s):  
Differential Equations ◽  
Machine Tool ◽  
Periodic Solutions ◽  
Stability Boundary ◽  
Spindle Speed ◽  
Regenerative Chatter ◽  
Machine Tool Chatter ◽  
Tool Motion ◽  
Nonlinear Delay ◽  
Tool Chatter

The mathematical models representing machine tool chatter dynamics have been cast as differential equations with delay. The suppression of regenerative chatter by spindle speed variation is attracting increasing attention. In this paper, we study nonlinear delay differential equations with periodic delays which models the machine tool chatter with continuously modulated spindle speed. The explicit time-dependent delay terms, due to spindle speed modulation, are replaced by state dependent delay terms by augmenting the original equations. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. We make use of Lyapunov-Schmidt Reduction method to determine the periodic solutions and analyze the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions close to the new stability boundary.

scholarly journals Numerical Stability Test of Neutral Delay Differential Equations

2008 ◽  
Vol 2008 ◽  
pp. 1-10 ◽  
Author(s):  
Z. H. Wang
Keyword(s):  
Characteristic Function ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Characteristic Root ◽  
Initial Guess ◽  
Lambert W Function ◽  
Neutral Delay ◽  
Neutral Delay Differential Equations ◽  
The Stability ◽  
Delay Differential

The stability of a delay differential equation can be investigated on the basis of the root location of the characteristic function. Though a number of stability criteria are available, they usually do not provide any information about the characteristic root with maximal real part, which is useful in justifying the stability and in understanding the system performances. Because the characteristic function is a transcendental function that has an infinite number of roots with no closed form, the roots can be found out numerically only. While some iterative methods work effectively in finding a root of a nonlinear equation for a properly chosen initial guess, they do not work in finding the rightmost root directly from the characteristic function. On the basis of Lambert W function, this paper presents an effective iterative algorithm for the calculation of the rightmost roots of neutral delay differential equations so that the stability of the delay equations can be determined directly, illustrated with two examples.

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