zbMATH — the first resource for mathematics

Semi-discretization method for delayed systems. (English) Zbl 1032.34071
The paper presents a numerical method for the stability analysis of autonomous or time periodic linear delayed systems. The method is based on a special kind of semidiscretization technique with respect to the delayed terms only. The resulting approximate system is also delayed and time periodic but can be transformed analytically into a finite-dimensional linear discrete system. The authors demonstrate their method by determining stability charts for the Mathieu equation with continuous time delay.

34K20 Stability theory of functional-differential equations
65L99 Numerical methods for ordinary differential equations
Full Text: DOI
[1] Stépán, Dynamics and Chaos in Manufacturing Processes pp 165– (1998)
[2] Stépán, Retarded Dynamical Systems (1989)
[3] Insperger T Stépán G Namachchivaya S Comparison of the dynamics of low immersion milling and cutting with varying spindle speed 2001
[4] Insperger, Stability of the milling process, Periodica Polytechnica 44 (1) pp 47– (2000)
[5] Insperger, Proceedings of 13th CISM-IFToMM Symposium on Theory and Practice of Robots and Manipulators pp 197– (2000)
[6] Routh, A Treatise on the Stability of a Given State of Motion (1877) · JFM 17.0315.02
[7] Hurwitz, Über die bedingungen unter welchen eine gleichung nur wurzeln mit negativen reellen teilen besitz, Mathematische Annalen 46 pp 74– (1895) · JFM 26.0119.03
[8] Floquet, Équations différentielles linéaires à coefficients périodiques, Annales Scientifiques de l’École Normale Supérieure 12 pp 47– (1883) · doi:10.24033/asens.220
[9] Hill, On the part of the motion of the lunar perigee which is a function of the mean motions of the Sun and Moon, Acta Mathematica 8 pp 1– (1886)
[10] D’Agelo, Linear Time-varying System: Analysis and Synthesis (1970)
[11] Nayfeh, Nonlinear Oscillations (1979)
[12] Sinha, An efficient computational scheme for the analysis of periodic systems, Journal of Sound and Vibration 151 pp 91– (1991) · doi:10.1016/0022-460X(91)90654-3
[13] van der Pol, On the stability of the solutions of Mathieu’s equation, Philosophical Magazine, and Journal of Science 5 pp 18– (1928) · JFM 54.0469.02
[14] Hale, Theory of Functional Differential Equations (1977) · Zbl 0352.34001
[15] Hale, Introduction to Functional Differential Equations (1993) · Zbl 0787.34002
[16] Diekmann, Delay Equations (1995)
[17] Bellman, Differential-Difference Equations (1963)
[18] Bhatt, Stability criteria for second-order dynamical systems with time lag, Journal of Applied Mechanics 33E (1) pp 113– (1966) · Zbl 0143.10503
[19] Neimark Ju, D-subdivision and spaces of quasi-polynomials, Prikladnaja Mathematika i Mechanika 13 (4) pp 349– (1949)
[20] Pontryagin, On the zeros of some elementary transcendental functions, Izvestiya Akademiya Nauk 6 (3) pp 115– (1942) · Zbl 0068.05803
[21] Kolmanovskii, Stability of Functional Differential Equations (1986)
[22] Hassard, Counting roots of the characteristic equation for linear delay-differential systems, Journal of Differential Equations 136 pp 222– (1997) · Zbl 0872.34051 · doi:10.1006/jdeq.1996.3127
[23] Farkas, Periodic Motions (1994)
[24] Balachandran, Non-linear dynamics of milling process, Philosophical Transactions of the Royal Society 359 pp 793– (2001) · Zbl 0961.74513
[25] Davies, Stability prediction for low radial immersion milling, Journal of Manufacturing Science and Engineering 124 (2) pp 217– (2002) · doi:10.1115/1.1455030
[26] Bayly PV Halley JE Mann BP Davies MA Stability of interrupted cutting by temporal finite element analysis 2001
[27] Seagalman, Suppression of regenerative chatter via impedance modulation, Journal of Vibration and Control 6 pp 243– (2000)
[28] Insperger T Stépán G Stability chart for the delayed mathieu equation 2002 · Zbl 1056.34073
[29] Insperger T Stépán G Semi-discretization of delayed dynamical systems 2001
[30] Hsu, Stability charts for second-order dynamical systems with time lag, Journal of Applied Mechanics 33E (1) pp 119– (1966) · Zbl 0143.10601
[31] Stépán, Vibrations of machines subjected to digital force control, International Journal of Solids and Structures 38 (10-13) pp 2149– (2001) · Zbl 0961.70507 · doi:10.1016/S0020-7683(00)00158-X
[32] Stépán, Varying delay and stability in dynamical systems, Zeitschrift für Angewandte Mathematik und Mechanik 71 (4) pp 154– (1991)
[33] Lakshmikantham, Theory of Difference Equations, Numerical Methods and Applications (1988) · Zbl 0683.39001
[34] Bauchau, An implicit floquet analysis for rotorcraft stability evaluation, Journal of the American Helicopter Society 46 pp 200– (2001)
[35] Farkas, On perturbation of the kernel in infinite delay systems, Zeitschrift für Angewandte Mathematik und Mechanik 72 (2) pp 153– (1992)
[36] Szabó, Parametric excitation of a single railway wheelset, Vehicle System Dynamics 33 (1) pp 49– (2000) · doi:10.1076/0042-3114(200001)33:1;1-5;FT049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.