A new analytic approach to obtain the complete solution for systems of delay differential equations (DDE) based on the concept of Lambert functions is presented. The similarity with the concept of the state transition matrix in linear ordinary differential equations enables the approach to be used for general classes of linear delay differential equations using the matrix form of DDEs. The solution is in the form of an infinite series of modes written in terms of Lambert functions. Stability criteria for the individual modes, free response, and forced response for delay equations in different examples are studied, and the results are presented. The new approach is applied to obtain the stability regions for the individual modes of the linearized chatter problem in turning. The results present a necessary condition to the stability in chatter for the whole system, since it only enables the study of the individual modes, and there are an infinite number of them that contribute to the stability of the system.
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June 2003
Technical Papers
Analysis of a System of Linear Delay Differential Equations
Farshid Maghami Asl, Graduate Student,
Farshid Maghami Asl, Graduate Student
Mechanical Engineering Department, University of Michigan, Ann Arbor, MI 48109-2125
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A. Galip Ulsoy, Professor
A. Galip Ulsoy, Professor
Mechanical Engineering Department, University of Michigan, Ann Arbor, MI 48109-2125
Search for other works by this author on:
Farshid Maghami Asl, Graduate Student
Mechanical Engineering Department, University of Michigan, Ann Arbor, MI 48109-2125
A. Galip Ulsoy, Professor
Mechanical Engineering Department, University of Michigan, Ann Arbor, MI 48109-2125
Contributed by the Dynamic Systems and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the ASME Dynamic Systems and Control Division, June 2002; final revision, January 2003. Associate Editor: N. Olgac.
J. Dyn. Sys., Meas., Control. Jun 2003, 125(2): 215-223 (9 pages)
Published Online: June 4, 2003
Article history
Received:
June 1, 2002
Revised:
January 1, 2003
Online:
June 4, 2003
Citation
Asl, F. M., and Ulsoy, A. G. (June 4, 2003). "Analysis of a System of Linear Delay Differential Equations ." ASME. J. Dyn. Sys., Meas., Control. June 2003; 125(2): 215–223. https://doi.org/10.1115/1.1568121
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