A simplified model of a container crane subject to a delayed feedback is investigated. The conditions for a Hopf bifurcation to stable/unstable limit-cycle solutions are determined. It is shown that a subcritical Hopf bifurcation to unstable oscillations cannot be ruled out and the undesired coexistence of stable large amplitude oscillations and a stable equilibrium endangers the robustness of time-delay control strategies. The bifurcation is analyzed both analytically and numerically using a continuation method.
Alsing, P.M., Kovanis, V., Gavrielides, A., and Erneux, T., 1996, “Lang and Kobayashi phase equation,” Physical Revue A 53, 4429—4434.
2.
Asme, 2005, 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Long Beach, CA, September.
3.
Campbell, S.A., Bélair, J., Ohira, T., and Milton, J., 1995, “Limit cycles, tori and complex dynamics in a second-order differential equation with delayed negative feedback,” Journal of Dynamics and Differential Equations7, 213—236.
4.
Casal, A. and Freedman, M., 1980, “A Poincaré—Lindstedt approach to bifurcation problems for differential-delay equations,” IEEE Transactions on Automatic Control25(5), 967—973.
5.
Drazin, P.G., 1992, Nonlinear Systems, Cambridge Texts in Applied Mathematics , Cambridge University Press, Cambridge .
6.
Engelborghs, K., Luzyanina, T., and Roose, D., 2002, “Numerical bifurcation analysis of delay differential equations using DDE-BIFTOOL,” ACM Transactions on Mathematics and Software28(1), 1—21.
7.
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.
8.
Erneux, T., 2005, “Multiple time scale analysis of delay differential equations modeling mechanical systems,” Proceedings of IDETC/CIE 2005, ASME 2005 International Design Engineering Technical Conferences, & Computers and Information in Engineering Conference , September, Long Beach, CA, USA.
9.
Gilsinn, D.E., 2002, “Estimating critical Hopf bifurcation parameters for a second-order delay differential equation with application to machine tool chatter,” Nonlinear Dynamics30(2), 103—154.
10.
Henry, R.J., 1999, Cargo pendulation reduction on ship-mounted cranes, Master's Thesis, Virginia Polytechnic Institute and State University , Blacksburg, VA.
11.
Henry, R.J., Masoud, Z.N., Nayfeh, A.H., and Mook, D.T., 2001, “Cargo pendulation reduction on ship-mounted cranes via boom-Lu angle actuation,” Journal of Vibration and Control7, 1253—1264.
12.
Kalmár-Nagy, T., Pratt, J.R., Davies, M.A., and Kennedy, M.D., 1999, “Experimental and analytical investigation of the subcritical instability in metal cutting,” in Proceedings of DETC'99 17th ASME Biennial Conferemce on Mechanical Vibration and Noise.
13.
Kalmár-Nagy, T., Stépán, G., and Moon, F.C., 2001, “Subcritical Hopf bifurcation in the delay equation model for machine tool vibrations,” Nonlinear Dynamics26, 121—142.
14.
Kalmár-Nagy, T., 2002, Delay-differential models of cutting tool dynamics with nonlinear and mode-coupling effects. PhD Thesis, Cornell University.
15.
Kevorkian, J. and Cole, J.D., 1981, Perturbation Methods in Applied Mathematics, Applied Mathematical Sciences 34, Springer, New York.
16.
Kevorkian, J. and Cole, J.D., 1996, Multiple Scale and Singular Perturbation Methods, Applied Mathematical Sciences 114, Springer, New York.
17.
Landry, M., Campell, S.A., Morris, K., and Aguilar, C.O., 2005, “Dynamics of an inverted pendulum with delayed feedback control,” SIAM Journal on Applied Dynamical Systems4(2), 333—351. http://www.math.uwaterloo.ca/~sacampbe/reprints/invpend.pdf
18.
Larger, L., Goedgebuer, J.-P., and Erneux, T., 2004, “Subcritical Hopf bifurcation in dynamical systems described by a scalar nonlinear delay differential equation,” Physical Revue E69, 036210.
19.
Masoud, Z.N., Nayfeh, A.H., and Al-Mousa, A., 2003, “Delayed position-feedback controller for the reduction of payload pendulations of rotary cranes,” Journal of Vibration and Control9, 257—277.
20.
Masoud, Z.N. and Nayfeh, A.H., 2003, “Sway reduction on container cranes using delayed feedback controller,” Nonlinear Dynamics34, 347—358.
21.
Masoud, Z.N., Nayfeh, A.H., and Mook, D.T., 2004, “Cargo pendulation reduction of ship-mounted cranes ,” Nonlinear Dynamics35, 299—311.
22.
Masoud, Z.N., Nayfeh, A.H., and Nayfeh, N.A., 2005, “Sway reduction on quay-side container cranes using delayed feedback controller: simulations and experiments,” Journal of Vibration and Control11, 1103— 1122.
23.
Minorsky, N., 1942, “Self-excited oscillations in dynamical systems possesing retarded actions,” Journal ofApplied Physics9, A65—A71
Nayfeh, N.A., 2002, Adaptation of delayed position feedback on the reduction of sway of container cranes. Master's thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.
26.
Nayfeh, N.A., Masoud, Z., and Baumann, W., 2005, “A comparison of three feedback controllers for container cranes,” in Proceedings of IDETC/CIE 2005, ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, September, Long Beach, CA, USA.
27.
Omar, H.M., 2003, Control of gantry and tower cranes. Master's thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.
28.
Pieroux, D., Erneux, T., and Otsuka, K., 1994, “Minimal model of a class-B laser with delayed feedback: cascading branching of periodic solutions and period-doubling bifurcation ,” Physical Revue A50, 1822—1829.
29.
Pieroux, D., Erneux, T., Gavrielides, A., and Kovanis, V., 2000, “Hopf bifurcation subject to a large delay in a laser system, ” SIAM Journal of Applied Mathematics61, 966—982); reprinted in SIAM Revue45, 523—540 (2003).
30.
Pyragas, K., 1992, “Continuous control of chaos by self-controlling feedback,” Physical Revue Letters A170, 421.
