Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter
In a turning process modeled using delay differential equations(DDEs), we investigate the stability of the regenerative machinetool chatter problem. An approach using the matrix Lambert Wfunction for the analytical solution to systems of delaydifferential equations is applied to this problem and com...
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AIMS Press
2007-01-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.355 |
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| author | Sun Yi Patrick W. Nelson A. Galip Ulsoy |
| author_facet | Sun Yi Patrick W. Nelson A. Galip Ulsoy |
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| collection | DOAJ |
| description | In a turning process modeled using delay differential equations(DDEs), we investigate the stability of the regenerative machinetool chatter problem. An approach using the matrix Lambert Wfunction for the analytical solution to systems of delaydifferential equations is applied to this problem and compared withthe result obtained using a bifurcation analysis. The Lambert Wfunction, known to be useful for solving scalar first-order DDEs,has recently been extended to a matrix Lambert W function approachto solve systems of DDEs. The essential advantages of the matrixLambert W approach are not only the similarity to the concept of thestate transition matrix in linear ordinary differential equations,enabling its use for general classes of linear delay differentialequations, but also the observation that we need only the principalbranch among an infinite number of roots to determine the stabilityof a system of DDEs. The bifurcation method combined with Sturmsequences provides an algorithm for determining the stability ofDDEs without restrictive geometric analysis. With this approach, onecan obtain the critical values of delay, which determine thestability of a system and hence the preferred operating spindlespeed without chatter. We apply both the matrix Lambert W functionand the bifurcation analysis approach to the problem of chatterstability in turning, and compare the results obtained to existingmethods. The two new approaches show excellent accuracy and certainother advantages, when compared to traditional graphical,computational and approximate methods. |
| format | Article |
| id | doaj-art-9aeb25bc929e420fafff8d14e2cd39d4 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2007-01-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-9aeb25bc929e420fafff8d14e2cd39d42025-01-24T01:53:28ZengAIMS PressMathematical Biosciences and Engineering1551-00182007-01-014235536810.3934/mbe.2007.4.355Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatterSun Yi0Patrick W. Nelson1A. Galip Ulsoy2Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125Department of Mechanical Engineering, University of Michigan, 2350 Hayward Street, Ann Arbor, MI 48109-2125In a turning process modeled using delay differential equations(DDEs), we investigate the stability of the regenerative machinetool chatter problem. An approach using the matrix Lambert Wfunction for the analytical solution to systems of delaydifferential equations is applied to this problem and compared withthe result obtained using a bifurcation analysis. The Lambert Wfunction, known to be useful for solving scalar first-order DDEs,has recently been extended to a matrix Lambert W function approachto solve systems of DDEs. The essential advantages of the matrixLambert W approach are not only the similarity to the concept of thestate transition matrix in linear ordinary differential equations,enabling its use for general classes of linear delay differentialequations, but also the observation that we need only the principalbranch among an infinite number of roots to determine the stabilityof a system of DDEs. The bifurcation method combined with Sturmsequences provides an algorithm for determining the stability ofDDEs without restrictive geometric analysis. With this approach, onecan obtain the critical values of delay, which determine thestability of a system and hence the preferred operating spindlespeed without chatter. We apply both the matrix Lambert W functionand the bifurcation analysis approach to the problem of chatterstability in turning, and compare the results obtained to existingmethods. The two new approaches show excellent accuracy and certainother advantages, when compared to traditional graphical,computational and approximate methods.https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.355lambert w functiondelay differential equationregenerative tool chatterbifurcation analysis. |
| spellingShingle | 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 Mathematical Biosciences and Engineering lambert w function delay differential equation regenerative tool chatter bifurcation analysis. |
| title | Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter |
| title_full | Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter |
| title_fullStr | Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter |
| title_full_unstemmed | Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter |
| title_short | Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter |
| title_sort | delay differential equations via the matrix lambert w function and bifurcation analysis application to machine tool chatter |
| topic | lambert w function delay differential equation regenerative tool chatter bifurcation analysis. |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2007.4.355 |
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