Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter
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Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125
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Department of Mathematics, University of Michigan, 525 East University, Ann Arbor, MI 48109-1109
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Received:
01 September 2006
Accepted:
29 June 2018
Published:
01 February 2007
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MSC :
34K20.
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In a turning process modeled using delay differential equations
(DDEs), we investigate the stability of the regenerative machine
tool chatter problem. An approach using the matrix Lambert W
function for the analytical solution to systems of delay
differential equations is applied to this problem and compared with
the result obtained using a bifurcation analysis. The Lambert W
function, known to be useful for solving scalar first-order DDEs,
has recently been extended to a matrix Lambert W function approach
to solve systems of DDEs. The essential advantages of the matrix
Lambert W approach are 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 W 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.
Citation: Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter[J]. Mathematical Biosciences and Engineering, 2007, 4(2): 355-368. doi: 10.3934/mbe.2007.4.355
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Abstract
In a turning process modeled using delay differential equations
(DDEs), we investigate the stability of the regenerative machine
tool chatter problem. An approach using the matrix Lambert W
function for the analytical solution to systems of delay
differential equations is applied to this problem and compared with
the result obtained using a bifurcation analysis. The Lambert W
function, known to be useful for solving scalar first-order DDEs,
has recently been extended to a matrix Lambert W function approach
to solve systems of DDEs. The essential advantages of the matrix
Lambert W approach are 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 W 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.
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