×

Stabilization via parametric excitation of multi-dof statically unstable systems. (English) Zbl 1470.70028

Summary: The problem of re-stabilization via parametric excitation of statically unstable linear Hamiltonian systems is addressed. An \(n\)-degree-of-freedom dynamical system is considered, at rest in a critical equilibrium position, possessing a pair of zero-eigenvalues and \(n-1\) pairs of distinct purely imaginary conjugate eigenvalues. The response of the system to a small static load, making the zero eigenvalues real and opposite, simultaneous to a harmonic parametric excitation of small amplitude, is studied by the Multiple Scale perturbation method, and the stability of the equilibrium position is investigated. Several cases of resonance between the excitation frequency and the natural non-zero frequencies are studied, calling for standard and non-standard applications of the method. It is found that the parametric excitation is able to re-stabilize the equilibrium for any value of the excitation frequencies, except for frequencies close to resonant values, provided a sufficiently large excitation amplitude is enforced. Results are compared with those provided by a purely numerical approach grounded on the Floquet theory.

MSC:

70K70 Systems with slow and fast motions for nonlinear problems in mechanics
70K28 Parametric resonances for nonlinear problems in mechanics
34D20 Stability of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Zelei, A.; Kovacs, L. L.; Stepan, G., Computed torque control of an under-actuated service robot platform modeled by natural coordinates, Commun Nonlinear Sci Numer Simul, 16, 5, 2205-2217 (2011) · Zbl 1221.70010
[2] Blajer, W.; Kolodziejczyk, K.; Mazur, Z., Inverse dynamics of underactuated mechanical systems: a simple case study and experimental verification, Commun Nonlinear Sci Numer Simul, 16, 5, 2265-2272 (2011) · Zbl 1221.70041
[3] Stepan, G.; Insperger, T., Stability of time-periodic and delayed systems - a route to act-and-wait control, IFAC Ann Rev Control, 30, 159-168 (2006)
[4] Insperger, T.; Wahi, P.; Colombo, A.; Stepan, G.; Di Bernardo, M.; Hogan, S. J., Full characterization of act-and-wait control for first-order unstable lag processes, J Vib Control, 16, 1209 (2010) · Zbl 1269.93034
[5] Levi, M., Geometry of vibrational stabilization and some applications, Int J Bifurcation Chaos, 15, 9, 2747-2756 (2005) · Zbl 1092.70513
[6] Champneys, A. R.; Fraser, W. B., The ‘Indian rope trick’ for a parametrically excited flexible rod: linearized analysis, Proc R Soc London A, 456, 553-570 (2000) · Zbl 0974.74029
[7] Mullin, T.; Champneys, A.; Fraser, W. B.; Galan, J.; Acheson, D., The Indian wire trick via parametric excitation: a comparison between theory and experiment, Proc R Soc London A, 459, 539-546 (2003) · Zbl 1047.70044
[8] Shishkina, E. V.; Blekhman, I. I.; Cartmell, M. P.; Gavrilov, S. N., Application of the method of direct separation of motions to the parametric stabilization of an elastic wire, Nonlinear Dyn, 54, 313-331 (2008) · Zbl 1173.74017
[9] Thomsen, J. J., Theories and experiments on the stiffening effect of high-frequency excitation for continuous elastic systems, J Sound Vib, 260, 117-139 (2003)
[10] Thomsen, J. J., Effective properties of mechanical systems under high-frequency excitation at multiple frequencies, J Sound Vib, 311, 1249-1270 (2008)
[11] Blekhman, I. I., Vibrational mechanics (2000), World Scientific: World Scientific New Jersey · Zbl 1046.74002
[12] Arkhipova, I. M.; Luongo, A.; Seyranian, A. P., Vibrational stabilization of the upright statically unstable position of a double pendulum, J Sound Vib, 331, 457-469 (2012)
[13] Seyranian, A. A.; Seyranian, A. P., The stability of an inverted pendulum with a vibrating suspension point, J Appl Math Mech, 70, 754-761 (2006) · Zbl 1126.70361
[14] Mailybaev, A. A.; Seyranian, A. P., Stabilization of statically unstable systems by parametric excitation, J Sound Vib, 323, 1016-1031 (2009)
[15] Nayfeh, A. H., Perturbation methods (1973), Wiley: Wiley New York · Zbl 0265.35002
[16] Luongo, A.; Di Egidio, A.; Paolone, A., Multiple time scale analysis for bifurcation from a multiple-zero eigenvalue, AIAA J, 41, 6, 1143-1150 (2003)
[17] Luongo, A.; Di Egidio, A.; Paolone, A., Multiscale analysis of defective multiple-Hopf bifurcations, Comput Struct, 82, 31-32, 2705-2722 (2004)
[18] Seyranian, A. P.; Mailybaev, A. A., Multiparameter stability theory with mechanical applications (2003), World Scientific: World Scientific New Jersey · Zbl 1047.34063
[19] Yakubovich, V. A.; Starzhinskii, V. M., Parametric resonance in linear systems (1987), Nauka: Nauka Moscow (in Russian) · Zbl 0623.34003
[20] Luongo, A.; Zulli, D., A paradigmatic system to study the transition from zero/Hopf to double-zero/Hopf bifurcation, Nonlinear Dyn, 70, 1, 111-124 (2012) · Zbl 1267.34067
[21] Luongo, A.; Paolone, A.; Di Egidio, A., Multiple time scales analysis for 1:2 and 1:3 resonant Hopf bifurcations, Nonlinear Dyn, 34, 3-4, 269-291 (2003) · Zbl 1041.70019
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.