Stability chart for the delayed Mathieu equation

In the space of system parameters, the closed–form stability chart is determined for the delayed Mathieu equation defined as ä(t)+(δ+ϵcost)x(t) = bx(t−2&pgr;). This stability chart makes the connection between the Strutt–Ince chart of the Mathieu equation and the Hsu–Bhatt–Vyshnegradskii chart of the second–order delay–differential equation. The combined chart describes the intriguing stability properties of a class of delayed oscillatory systems subjected to parametric excitation.

[1]  Gábor Stépán,et al.  Remote Control of Periodic Robot Motion , 2000 .

[2]  B. Mann,et al.  Stability of Interrupted Cutting by Temporal Finite Element Analysis , 2003 .

[3]  C. Hsu,et al.  Stability Criteria for Second-Order Dynamical Systems With Time Lag , 1966 .

[4]  J. Hale Theory of Functional Differential Equations , 1977 .

[5]  V. Kolmanovskii,et al.  Stability of Functional Differential Equations , 1986 .

[6]  Martin Odersky,et al.  An Introduction to Functional Nets , 2000, APPSEM.

[7]  Jon R. Pratt,et al.  Stability Prediction for Low Radial Immersion Milling , 2002 .

[8]  Henry D'Angelo,et al.  Linear time-varying systems : analysis and synthesis , 1970 .

[9]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[10]  Balakumar Balachandran,et al.  Nonlinear dynamics of milling processes , 2001, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[11]  Miklós Farkas,et al.  Periodic Motions , 1994 .

[12]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[13]  Balth van der Pol Jun. Doct.Sc. II. On the stability of the solutions of Mathieu's equation , 1928 .

[14]  D. Segalman,et al.  Suppression of Regenerative Chatter via Impedance Modulation , 2000 .

[15]  L. Rayleigh,et al.  XVII. On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure , 1887 .

[16]  E. Mathieu Mémoire sur le mouvement vibratoire d'une membrane de forme elliptique. , 1868 .

[17]  S. Sinha,et al.  An efficient computational scheme for the analysis of periodic systems , 1991 .

[18]  G. Stépán,et al.  Stability of the milling process , 2000 .

[19]  G. Hill On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and moon , 1886 .