In this paper, a time-delayed positive position feedback controller is proposed to reduce the horizontal vibration of a magnetically levitated body subjected to multi force excitations. This controller is coupled to the main system with 1:1 internal resonance. The method of multiple scales perturbation technique is employed to obtain an approximate solution which clarifies the nonlinear behavior for both amplitude and phase of the whole system. The effects of time delay magnitude are investigated to indicate the safe region of operation. Validation curves are involved to compare the approximate solution with the numerical simulation.
AhlbornAParlitzU (2005) Controlling dynamical systems using multiple delay feedback control. Physical Review E72: 016206.
2.
AmerYABauomyHSSayedM (2009) Vibration suppression in a twin-tail system to parametric and external excitation. Computers and Mathematics with Applications58: 1947–1964.
3.
DasSLChatterjeeA (2002) Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonlinear Dynamics30: 323–335.
4.
El-GanainiWAAEl-GoharyHA (2011) Vibration suppression via time-delay absorber described by non-linear differential equations. Advances in Theoretical and Applied Mechanics4(2): 49–67.
5.
El-GoharyHAEl-GanainiWAA (2012) Vibration suppression of a dynamical system to multi-parametric excitations via time delay absorber. Applied Mathematical Modelling36: 35–45.
6.
El-GanainiWASaeedNAEissaM (2013) Positive position feedback controller (PPF) for suppression of nonlinear sytem vibration. Nonlinear Dynamics72: 517–537.
7.
EissaMSayedM (2006a) A comparison between active and passive vibration control of non-linear simple pendulum, Part I: transversally tuned absorber and negative feedback. Mathematical and Computational Applications11(2): 137–149.
8.
EissaMSayedM (2006b) A comparison between active and passive vibration control of non-linear simple pendulum, Part II: longitudinal tuned absorber and negative and feedback. Mathematical and Computational Applications11(2): 151–162.
9.
EissaMSayedM (2008) Vibration reduction of a three DOF non-linear spring pendulum. Communications in Nonlinear Science and Numerical Simulation13: 465–488.
10.
EissaMBauomyHSAmerYA (2007) Active control of an aircraft tail subject to harmonic excitation. Acta Mechanica Sinica23: 451–462.
11.
EissaMEl-GanainiWAAHamedYS (2005) Saturation, stability and resonance of non-linear systems. Physica A356: 341–358.
12.
GjurchinovskiAJunglingTUrumovVSchollE (2013) Delayed feedback control of unstable steady states with high-frequency modulation of the delay. Physical Review E88: 032912.
13.
JoHYabunoH (2009) Amplitude reduction of primary resonance of nonlinear oscillator by a dynamic vibration absorber using nonlinear coupling. Nonlinear Dynamics55: 67–78.
14.
JoHYabunoH (2010) Amplitude reduction of parametric resonance by dynamic vibration absorber based on quadratic nonlinear coupling. Journal of Sound and Vibration329: 2205–2217.
15.
NayfehAHMookDT (1995) Nonlinear Oscillations, New York: Wiley.
16.
SaeedNAEl-GaniniWAEissaM (2013) Nonlinear time delay saturation-based controller for suppression of nonlinear beam vibrations. Applied Mathematical Modelling37: 8846–8864.
17.
SayedMKamelM (2012) 1:2 and 1:3 internal resonance active absorber for non-linear vibrating system. Applied Mathematical Modelling36: 310–332.
18.
ShinCHongCJeongWB (2012) Active vibration control of clamped beams using positive position feedback controllers with moment pair. Journal of Mechanical Science and Technology26(3): 731–740.
19.
WarminskiJBochenskiMJarzynaWFilipekPAugustyniakM (2011) Active suppression of nonlinear composite beam vibrations by selected control algorithms. Communications in Nonlinear Science Numerical Simulation16: 2237–2248.
20.
XuJChungKWZhaoYY (2010) Delayed saturation controller for vibration suppression in a stainless-steel beam. Nonlinear Dynamics62: 177–193.
21.
YabunoHJoH (2008) Parametric resonance due to asymmetric nonlinearity of restoring force. Journal of System Design and Dynamics2(3): 898–907.
22.
ZhaoYYXuJ (2007) Effects of delayed feedback control on nonlinear vibration absorber system. Journal of Sound and Vibration308: 212–230.
23.
ZhaoYYXuJ (2012) Using the delayed feedback control and saturation control to suppress the vibration of the dynamical system. Nonlinear Dynamics67: 735–753.
