In the space of the system parameters, the stability charts are determined for the delayed and damped Mathieu equation defined as This stability chart makes the connection between the Strutt-Ince chart of the damped Mathieu equation and the Hsu-Bhatt-Vyshnegradskii chart of the autonomous second order delay-differential equation. The combined charts describe the intriguing stability properties of an important class of delayed oscillatory systems subjected to parametric excitation.
Issue Section:
Technical Papers
1.
van de Pol
, F.
, and Strutt
, M. J. O.
, 1928
, “On the Stability of the Solutions of Mathieu’s Equation
,” Philos. Mag.
, 5
, pp. 18
–38
.2.
Hsu
, C. S.
, and Bhatt
, S. J.
, 1966
, “Stability Charts for Second-Order Dynamical Systems With Time Lag
,” ASME J. Appl. Mech.
, 33E
(1
), pp. 119
–124
.3.
Insperger
, T.
, and Ste´pa´n
, G.
, 2002
, “Stability Chart for the Delayed Mathieu Equation
,” Proc. R. Soc., Math. Physic. Eng. Sci.
, 458
(2024
), pp. 1989
–1998
.4.
Floquet
, M. G.
, 1883
, “E´quations Diffe´rentielles Line´aires a Coefficients Pe´ridiques
,” Ann. Sci. Ec. Normale Super.
, 12
, pp. 47
–89
.5.
Farkas, M., 1994, Periodic Motions, Springer-Verlag, New York.
6.
Hill
, G. W.
, 1886
, “On the Part of the Motion of the Lunar Perigee Which is a Function of the Mean Motions of the Sun and Moon
,” Acta Math.
, 8
, pp. 1
–36
.7.
Rayleigh (Strutt
), J. W.
, 1887
, “On the Maintenance of Vibrations by Forces of Double Frequency, and on the Propagation of Waves Through a Medium Endowed With a Periodic Structure
,” Philos. Mag.
, 24
, pp. 145
–159
.8.
D’Agelo, H., 1970, Linear Time-Varying System: Analysis and Synthesis, Allyn and Bacon, Boston.
9.
Insperger
, T.
, and Horva´th
, R.
, 2000
, “Pendulum with Harmonic Variation of the Suspension Point
,” Period. Polytech., Mech. Eng.-Masinostr.
, 44
(1
), pp. 39
–46
.10.
Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, John Wiley and Sons, New York.
11.
Sinha
, S. C.
, and Wu
, D. H.
, 1991
, “An Efficient Computational Scheme for the Analysis of Periodic Systems
,” J. Sound Vib.
, 151
, pp. 91
–117
.12.
Sinha
, S. C.
, and Butcher
, E. A.
, 1997
, “Symbolic Computation of Fundamental Solution Matrices for Linear Time-Periodic Dynamical Systems
,” J. Sound Vib.
, 206
, pp. 61
–85
.13.
Bauchau
, O. A.
, and Nikishkov
, Y. G.
, 2001
, “An Implicit Floquet Analysis for Rotorcraft Stability Evaluation
,” J. Am. Helicopter Soc.
, 46
, pp. 200
–209
.14.
Mathieu
, E.
, 1868, “Me´moire sur le Mouvement Vibratoire d’une Membrane de Forme Elliptique,” J. Math., 13, pp. 137–203.15.
Stephenson, A., 1908, “On a New Type of Dynamical Stability,” Mem. Proc. Manch. Lit. Phil. Soc., 52, pp. 1–10.
16.
Volterra
, V.
, 1928
, “Sur la Theorie Mathematique des Phenomenes Hereditares
,” J. Math. Pures Appl.
, 7
, pp. 149
–192
.17.
von Schlippe, B., and Dietrich, R., 1941, “Shimmying of a Pneumatic Wheel,” Lilienthal-Gesellschaft fu¨r Luftfahrtforschung, Bericht, 140 (Translated for AAF in 1947 by Meyer & Co., pp. 125–160).
18.
Minorsky
, N.
, 1942
, “Selfexcited Oscillations in Dynamical Systems Possessing Retarded Actions
,” ASME J. Appl. Mech.
, 9
, pp. 65
–71
.19.
Tlusty, J., Polacek, A., Danek, C., and Spacek, J., 1962, Selbsterregte Schwingungen an Werkzeugmaschinen, VEB Verlag Technik, Berlin.
20.
Tobias, S. A., 1965, Machine Tool Vibration, Blackie, London.
21.
Kudinov, V. A., 1955, “Theory of Vibration Generated from Metal Cutting” (in Russian), New Technology of Mechanical Engineering, USSR Academy of Sciences Publishing House, Moscow, pp. 1–7.
22.
Kudinov, V. A., 1967, Dynamics of Tool-Lathe (in Russian), Mashinostroenie, Moscow.
23.
Ste´pa´n, G., 1989, Retarded Dynamical Systems, Longman, Harlow.
24.
Moon, F. C., 1998, Dynamics and Chaos in Manufacturing Processes, Wiley, New York.
25.
Whitney
, D. E.
, 1977
, “Force Feedback Control of Manipulator Fine Motions
,” ASME J. Dyn. Syst., Meas., Control
, 98
, pp. 91
–97
.26.
Ste´pa´n, G., and Steven, A., 1990, “Theoretical and Experimental Stability Analysis of a Hybrid Position-Force Controlled Robot,” Proc. of 8th Symp. on Theory and Practice of Robots and Manipulators, Krakow, Poland, pp. 53–60.
27.
Kim
, W. S.
, and Bejczy
, A. K.
, 1993
, “Demonstration of a High-Fidelity Predictive/Preview Display Technique for Telerobotic Servicing in Space
,” IEEE Trans. Rob. Autom.
, 9
(5
), pp. 698
–702
.28.
Campbell
, S. A.
, Ruan
, S.
, and Wei
, J.
, 1999
, “Qualitative Analysis of a Neural Network Model With Multiple Time Delays
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
, 9
(8
), pp. 1585
–1595
.29.
Myshkis
, A. D.
, 1949, “General Theory of Differential Equations with Delay,” Uspehi Mat. Nauk., 4(5), pp. 99–141 (Engl. Transl., AMS 1951, 55, pp. 1–62.30.
Myshkis, A. D., 1955, Lineare Differentialgleichungen mit nacheilendem Argument, Deutscher Verlag der Wissenschaften, Berlin.
31.
Bellman, R., and Cooke, K., 1963, Differential-Difference Equations, Academic Press, New York.
32.
Halanay, A., 1966, Differential Equations: Stability, Oscillations, Time Lags, Academic Press, New York.
33.
Hale, J. K., 1977, Theory of Functional Differential Equations, Springer-Verlag, New York.
34.
Kolmanovskii, V. B., and Nosov, V. R., 1986, Stability of Functional Differential Equations, Academic Press, London.
35.
Hale, J. K., and Lunel, S. M. V., 1993, Introduction to Functional Differential Equations, Springer-Verlag, New York.
36.
Diekmann, O., van Gils, S. A., Lunel, S. M. V., and Walther, H.-O., 1995, Delay Equations, Springer-Verlag, New York.
37.
Bhatt
, S. J.
, and Hsu
, C. S.
, 1966
, “Stability Criteria for Second-Order Dynamical Systems With Time Lag
,” ASME J. Appl. Mech.
, 33E
(1
), pp. 113
–118
.38.
Neimark
, Ju. I.
, 1949
, “D-subdivision and Spaces of Quasi-Polynomials
” (in Russian), Prikl. Mat. Mekh.
, 13
(4
), pp. 349
–380
.39.
Pontryagin
, L. S.
, 1942
, “On the Zeros of Some Elementary Transcendental Functions,” (in Russian
), Izv. Akad. Nauk SSSR
, 6
(3
), pp. 115
–134
.40.
Olgac
, N.
, and Sipahi
, R.
, 2002
, “An Exact Method for the Stability Analysis of Time Delayed LTI Systems
,” IEEE Trans. Autom. Control
, 47
(5
), pp. 793
–797
.41.
Halanay
, A.
, 1961
, “Stability Theory of Linear Periodic Systems With Delay” (in Russian
), Rev. Math. Pures Appl.
, 6
(4
), pp. 633
–653
.42.
Seagalman
, D. J.
, and Butcher
, E. A.
, 2000
, “Suppression of Regenerative Chatter via Impendance Modulation
,” J. Vib. Control
, 6
, pp. 243
–256
.43.
Insperger
, T.
, and Ste´pa´n
, G.
, 2000
, “Stability of the Milling Process
,” Period. Polytech. Mech. Eng.-Masinostr.
, 44
(1
), pp. 47
–57
.44.
Bayly, P. V., Halley, J. E., Mann, B. P., and Davies, M. A., 2001, “Stability of Interrupted Cutting by Temporal Finite Element Analysis,” Proc. of the ASME 2001 Design Engineering Technical Conf., Pittsburgh, PA, Paper No. DETC2001/VIB-21581 (CD-ROM).
45.
Insperger
, T.
, and Ste´pa´n
, G.
, 2002
, “Semi-Discretization Method for Delayed Systems
,” Int. J. Numer. Methods Eng.
, 55
(5
), pp. 503
–518
.46.
Zhao
, M. X.
, and Balachandran
, B.
, 2001
, “Dynamics and Stability of Milling Process
,” Int. J. Solids Struct.
, 38
(10–13
), pp. 2233
–2248
.47.
Schmitz
, T. L.
, Davies
, M. A.
, Medicus
, K.
, and Snyder
, J.
, 2001
, “Improving High-Speed Machining Material Removal Rates by Rapid Dynamic Analysis
,” Annals of the CIRP
, 50
(1
), pp. 263
–268
.48.
Mennicken
, R.
, 1968
, “On the Convergence of Infinite Hill-Type Determinants
,” Arch. Ration. Mech. Anal.
, 30
, pp. 12
–37
.49.
Denk
, R.
, 1995
, “Hill’s Equation Systems and Infinite Determinants
,” Math. Nachr.
, 175
, pp. 47
–60
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