Abstract
This paper considers the effect of time delays on the saturation control of first-mode vibration of a stainless-steel beam. Time delay is commonly caused by measurements of the system states, transport delay, on-line computation, filtering and processing of data, calculating and executing of control forces as required in control processing. The method of multiple scales is employed to obtain the analytical solutions of limit cycles and their stability and to investigate the bifurcations of the system under consideration. All the predictions from analytical solutions are in agreement with the numerical simulation. The analytical results show that a delay can change the range of the saturation control, either widening or shrinking the effective frequency bandwidth. Thus, vibration control of a beam can be achieved using an appropriate choice of the delay in a self-feedback signal. From the examples illustrated, this paper provides a positive example that time delay can also be utilized to suppress vibration in systems when time delay cannot be neglected.
Similar content being viewed by others
References
Haxton, R., Barr, A.: The autoparametric vibration absorber. J. Eng. Ind. 94, 119–225 (1972)
Nayfeh, A.H., Mook, D.T., Marshall, L.R.: Nonlinear coupling of pitch and roll modes in ship motion. J. Hydronaut. 7, 145–152 (1973)
Haddow, A.G., Barr, A.D.S., Mook, D.T.: Theoretical and experimental study of modal interaction in a two degree-of-freedom structure. J. Sound Vib. 97(3), 451–473 (1984)
Nayfeh, A.H., Zavodney, L.D.: Experimental observation of amplitude- and phase-modulated responses of two internally coupled oscillators to a harmonic excitation. ASME J. Appl. Mech. 55, 706–710 (1988)
Nayfeh, A.H., Balachandran, B.: Modal interactions in dynamical and structural systems. ASME Appl. Mech. Rev. 42(11), 175–201 (1989)
Oueini, S.S., Nayfeh, A.H., Golnaraghi, M.F.: A theoretical and experimental implementation of a control method based on saturation. Nonlinear Dyn. 13, 189–202 (1997)
Oueini, S.S., Golnaraghi, M.F.: Experimental implementation of the internal resonance control strategy. J. Sound Vib. 191, 377–396 (1996)
Oueini, S.S., Nayfeh, A.H.: Analysis and application of a nonlinear vibration absorber. J. Vib. Control 6(7), 999–1016 (2000)
Pai, P.F., Wen, B., Naser, A.S., Schultz, M.J.: Structural vibration control using PZT patches and non-linear phenomena. J. Sound Vib. 215, 273–296 (1998)
Pratt, J.R., Oueini, S.S., Nayfeh, A.H.: A Terfenol-D nonlinear vibration absorber. J. Intell. Mater. Syst. Struct. 10, 29–35 (1999)
Pai, P.F., Wen, B., Naser, A.S., Schultz, M.J.: Nonlinear vibration suppression of cantilever beams using bi-moments induced by PZT actuators. In: Proceedings of 38th AIAA Structures, Structural Dynamics & Materials Conference, Hyatt Orlando, FL, pp. 1789–1795 (1997)
Pai, P.F., Schultz, M.J.: A refined nonlinear vibration absorber. Int. J. Mech. Sci. 42, 537–560 (2000)
Saguranrum, S., Kunz, D.L., Omar, H.M.: Numerical simulations of cantilever beam response with saturation control and full modal coupling. Comput. Struct. 81, 1499–1510 (2003)
Shoeybi, M., Ghorashi, M.: Control of a nonlinear system using the saturation phenomenon. Nonlinear Dyn. 42, 113–126 (2005)
Shoeybi, M., Ghorashi, M.: Nonlinear vibration control of a system with dry friction and viscous damping using the saturation phenomenon. Nonlinear Dyn. 45, 249–272 (2006)
Li, J., Hua, H.X., Shen, R.Y.: Saturation-based active absorber for a non-linear plant to a principal external excitation. Mech. Syst. Signal Process. 21, 1489–1498 (2007)
Chu, S.Y., Soong, T.T., Lin, C.C., Chen, Y.Z.: Time-delay effect and compensation on direct output feedback controlled mass damper system. Earthquake Eng. Struct. Dyn. 32, 121–137 (2002)
Yang, B., Mote, C.D.: Noncollocated control of a damped string using time delay. J. Dyn. Syst. Meas. Control 114(4), 736–740 (1992)
Abdel, M.M., Roorda, J.: Time delay compensation in active damping of structures. J. Eng. Mech. 117(11), 2549–2570 (1991)
Rodellar, J., Chung, L.L., Soong, T.T., Reinhorn, A.M.: Experimental digital control of structures. J. Eng. Mech. 115, 1245–1261 (1989)
Xu, J., Chung, K.W.: Effects of time delayed position feedback on a van der Pol–Duffing oscillator. Physica D 180 (1–2), 17–39 (2003)
Guan, Y.H., Lim, T.C., Shepard, W.S.: Experimental study on active vibration control of a gearbox system. J. Sound Vib. 282, 713–733 (2005)
Hu, H.Y., Dowell, E.H., Virgin, L.: Resonances of a harmonically forced Duffing oscillator with time delay state feedback. Nonlinear Dyn. 45, 249–272 (2006)
Xu, J., Yu, P.: Delay-induced bifurcations in a nonautonomous system with delayed velocity feedbacks. Int. J. Bifurc. Chaos 14(8), 2777–2798 (2004)
Olgac, N., McFarland, D., Holm-Hansen, B.T.: Position feedback-induced resonance: the delayed resonator. In: ASME Winter Annu. Meet., vol. 38, pp. 113–119 (1992)
Filipovic, D., Olgac, N.: Delayed resonator with speed feedback—design and performance analysis. Mechatronics 12(3), 393–413 (2002)
Zhao, Y.Y., Xu, J.: Effects of delayed feedback control on nonlinear vibration absorber system. J. Sound Vib. 308, 212–230 (2007)
Wirkus, S., Rand, R.: The dynamics of two coupled van der Pol oscillators with delay coupling. Nonlinear Dyn. 30, 205–221 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xu, J., Chung, K.W. & Zhao, Y.Y. Delayed saturation controller for vibration suppression in a stainless-steel beam. Nonlinear Dyn 62, 177–193 (2010). https://doi.org/10.1007/s11071-010-9708-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-010-9708-4