High-dimensional harmonic balance analysis for second-order delay-differential equations

JVC/Journal of Vibration and Control

Volumn 16, Issue 7-8, 2010, Pages 1189-1208

  Liu, Liping ^a   Kalmár Nagy, Tamás ^b  

^a NORTH CAROLINA A AND T STATE UNIVERSITY   (United States)

^b Texas AandM University   (United States)

Author keywords

Circadian rhythm; delay differential equations; harmonic balance; Hopf bifurcation; machine tool vibrations; sunflower equation

Indexed keywords

CIRCADIAN RHYTHMS; DELAY DIFFERENTIAL EQUATIONS; HARMONIC BALANCE; HARMONIC BALANCE ANALYSIS; HARMONIC BALANCE METHOD; HIGH-DIMENSIONAL; LIMIT CYCLE; MACHINE TOOL VIBRATIONS; MATRIX; SECOND ORDERS; UNSTABLE LIMIT CYCLES;

DIFFERENTIAL EQUATIONS; HARMONIC ANALYSIS; MACHINE TOOLS;

HOPF BIFURCATION;

EID: 77954333728     PISSN: 10775463     EISSN: 17412986     Source Type: Journal    
DOI: 10.1177/1077546309341134     Document Type: Article

References (51)

1. Asl, F.M. and Ulsoy, A.G., 2003, " Analysis of a system of linear delay differential equations, " Journal of Dynamic Systems, Measurement, and Control 125, 215-223.
2. Beuter, A., Bélair, J., and Labrie, C., 1993, " Feedback and delays in neurological disease: a modeling study using dynamical systems, " Bulletin of Mathematical Biology 55, 525-541.
3. Breda, D., Maset, S., and Vermiglio, R., 2004, " Computing the characteristic roots for delay differential equations, " IMA Journal of Numerical Analysis 24, 1-19.
4. Butcher, E.A., Ma, H., Bueler, E., Averina, V., and Szabó, Zs., 2004, " Stability of time-periodic delay-differential equations via Chebyshev polynomials, " International Journal for Numerical Methods in Engineering 59, 895-922.
5. Casal, A. and Freedman, M., 1980, " A Poincaré-Lindstedt approach to bifurcation problems for differential delay equations, " IEEE Transactions on Automatic Control 25 (5). 967-973.
6. Casal, A. and Somolinos, A., 1982, " Forced oscillations for the sunflower equation, entrainment, " Nonlinear Analysis Theory, Methods and Applications 6 (4). 397-414.
7. Dowell, E.H. and Hall, K.C., 2001, " Modeling of fluid-structure interaction, " Annual Review of Fluid Mechanics 33, 413-455.
8. Engelborghs, K., Luzyanina, T., and Roose, D., 2002, " Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL, " ACM Transactions on Mathematical Software 28 (1). 1-21. (Pubitemid 135686769)
9. Engelborghs, K., Luzyanina, T., and Samaey, G., 2001, " DDE-BIFTOOL v. 2.00: a Matlab package for bifurcation analysis of delay differential equations, " Technical Report TW-330, Department of Computer Science, K.U. Leuven, Leuven, Belgium.
10. Erneux, T. and Kalmár-Nagy, T., 2007, " Nonlinear stability of a delayed feedback controlled container crane, " Journal of Vibration and Control 13 (5). 603.
11. Gilsinn, D.E., 2005, " Discrete Fourier series approximation to periodic solutions of autonomous delay differential equations, " in Proceedings of IDERC/CIE 2005: ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Long Beach, CA, September 24-28.
12. Hall, K.C., Thomas, J.P., and Clark, W.S., 2000, " Computation of unsteady nonlinear flows in cascades using a harmonic balance technique, " in 9th International Symposium on Unsteady Aerodynamics, Aeroacoustics and Aeroelasticity of Turbomachines (ISUAAAT), Lyon, France, September.
13. Hassard, B.D., Kazarinoff, N.D., and Wan Y.-H., 1981, Theory and Applications of Hopf Bifurcation (London Mathematical Society Lecture Note Series, Vol. 41), Cambridge University Press, Cambridge.
14. Hu, H.Y., Dowell, E.H., and Virgin, L.N., 1998, " Resonance of a harmonically forced Duffing oscillator with time delay state feedback, " Nonlinear Dynamics 15 (4). 311-327.
15. Hu, H.Y. and Wang, Z.H., 2002, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer, Berlin.
16. Insperger T. and Stépán, G., 2002, " Semidiscretization method for general delayed systems, " International Journal for Numerical Methods in Engineering 55, 503-518.
17. Insperger, T., Wahi, P., Colombo, A., Stépán, G., di Bernardo, M., and Hogan, S.J., 2010, "Full characterization of act-and-wait control for first order unstable lag processes," Journal of Vibration and Control (in press).
18. Israelsson, D. and Johnsson, A., 1967, " A theory for circumnutations in Helianthus annuus, " Physiologia Plantarum 20, 957-976.
19. Kalmár-Nagy, T., 2005, " A new look at the stability analysis of delay-differential equations, " in Proceedings of the DETC, Long Beach, CA, September 24-28.
20. Kalmár-Nagy, T., Pratt, J.R., Davies, M.A., and Kennedy, M.D., 1999, " Experimental and analytical investigation of the subcritical instability in turning, " in Proceedings of the 1999 ASME Design Engineering Technical Conferences, 17th ASME Biennial Conference on Mechanical Vibration and Noise, Las Vegas, NV, DETC99/VIB-8060.
21. Kalmár-Nagy, T., Stépán G., and Moon, F.C., 2001, " Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations, " Nonlinear Dynamics 26, 121-142.
22. Kim, Y.B., Noah S.T., and Choi, Y.S., 1991, " Periodic response of multi-disk rotors with bearing clearances, " Journal of Sound and Vibrations 144 (3). 381.
23. Krasnolselskii, M.A., 1984, " On the problem of forced oscillation in a nonlinear control systems, " Engineering Cybernetics 22, 92-99.
24. Kuang, Y., 1993, Delay Differential Equations with Applications to Population Dynamics, Academic Press, New York.
25. Liu, L. and Dowell, E.H., 2005, " Harmonic balance approach for an airfoil with a freeplay control surface, " AIAA Journal 43 (4). 802.
26. Liu, L., Thomas, J.P., Dowell, E.H., Attar, P., and Hall, K.C., 2006, " A comparison of classical and high dimensional harmonic balance approaches for a Duffing oscillator, " Journal of Computational Physics 215 (1). 298-320. (Pubitemid 43574493)
27. MacDonald, N., 1989, Biological Delay Systems: Linear Stability Theory, Cambridge University Press, New York.
28. MacDonald, N., 1995, " Harmonic balance in delay-differential equations, " Journal of Sound and Vibration 186 (4). 649-656.
29. Mann, B.P. and Patel, B.R., 2010, "Stability of delay equations written as state space models," Journal of Vibration and Control (in press).
30. McMullen, M., Jameson, A., and Alonso, J.J., 2001, " Acceleration of convergence to a periodic steady state in turbomachinery flows, " AIAA Paper 2001-0152.
31. Mickens, R.E., 1996, Oscillations in Planar Dynamic Systems (Advances in Mathematics for Applied Sciences, Vol. 37), World Scientific, River Edge, New Jersey.
32. Morris, H.C., 1976, " A perturbative approach to periodic solutions of delay differential equations, " IMA Journal of Applied Mathematics 18 (1). 15-24.
33. Ohlsson, H., 2006, " Mathematical analysis of a biological clock model, " Masters dissertation, Linkopings University, Sweden.
34. Olgac, N. and Sipahi, R., 2002, " An exact method for the stability analysis of time-delayed LTI systems, " IEEE Transactions on Automatic Control 47 (5). 793-797.
35. Pieroux, D., Erneux, T., Luzyanina, T., and Engelborghs, K., 2001, " Interacting pairs of periodic solutions lead to tori in lasers subject to delayed feedback, " Physical Review E 63 (3). 36211.
36. Pyragas, K., 1992, " Continuous control of chaos by self-controlling feedback, " Physical Review Letters 78 (2). 421-428.
37. Ragulskis, M., Fedaravicius, A., and Ragulskis K., 2006, " Harmonic balance method for FEM analysis of fluid flow in a vibrating pipe, " Communications in Numerical Methods in Engineering 22 (5). 347-356.
38. Saupe, D., 1983, " Global bifurcation of periodic solutions of some autonomous differential delay equations, " Applied Mathematics and Computing 13, 185-211.
39. Shayer, L. and Campbell, S.A., 2000, " Stability, bifurcation, and multi-stability in a system of two coupled neurons with multiple time delays, " SIAM Journal of Applied Mathematics 61, 673-700.
40. Stépán, G., 1989, Retarded Dynamical Systems: Stability and Characteristic Functions (Pitman Research Notes in Mathematics, Vol. 210), Longman Scientific and Technical, Harlow.
41. Stépán, G., 1997, " Delay-differential equation models for machine tool chatter, " in F. C. Moon (ed.), Dynamics and Chaos in Manufacturing Processes, Wiley, New York, pp. 165-191.
42. Stépán, G., 2000, " Modeling nonlinear regenerative effects in metal cutting, " Philosophical Transactions of the Royal Society of London A 359, 739-757.
43. Szalai, R., Stépán, G., and Hogan, S.J., 2006, " Continuation of bifurcations in periodic delay-differential equations using characteristic matrices, " SIAM Journal on Scientific Computing 28 (4). 1301-1317.
44. Thomas, J.P., Dowell, E.H., and Hall, K.C., 2002 a, " Nonlinear inviscid aerodynamic effects on transonic divergence, flutter, and limit-cycle oscillations " AIAA Journal 40 (4). 638-646.
45. Thomas, J.P., Dowell, E.H., and Hall, K.C., 2002 b, " Modeling viscous transonic limit cycle oscillation behavior using a harmonic balance approach, " AIAA Paper 2002-1414.
46. Thomas, J.P., Hall, K.C., and Dowell, E.H., 2003, " A harmonic balance approach for modeling nonlinear aeroelastic bahvior of wings in transonic flow, " AIAA Paper 2003-1924.
47. Verdugo, A. and Rand, R., 2008, " Hopf bifurcation in a DDE model of gene expression, " Communications in Nonlinear Science and Numerical Simulation 13, 235-242. (Pubitemid 47289299)
48. Wahi, P. and Chatterjee, A., 2005, " Galerkin projections for delay differential equations, " Journal of Dynamic Systems, Measurement, and Control 127 (1). 80-87.
49. Wang, H. and Hu, H.Y., 2003, " Remarks on the perturbation methods in solving the second order delay differential equations, " Nonlinear Dynamics 33, 379-398.
50. Wu, D.M. and Wang, Z., 2006, " A Mathematica program for the approximate analytical solution to a nonlinear undamped Duffing equation by a new approximate approach, " Computer Physics Communications 174 (6). 447-463.
51. Yi, S., Ulsoy, A.G., and Nelson, P.W., 2006, " Analysis of systems of linear delay differential equations using the matrix Lambert function and the Laplace transformation, " in Proceedings of CDC 2006, San Diego, CA.