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Resonances of a harmonically forced Duffing oscillator with time delay state feedback. (English) Zbl 0906.34052
Authors’ summary: “The paper presents analytical and numerical studies of primary resonance and \(1/3\) subharmonic resonance of a forced Duffing oscillator under state feedback control with a time delay. By using the method of multiple scales, a first-order approximation to the resonances are derived and the effect of time delay on the resonances is analysed. The concept of an equivalent damping related to the delay feedback is proposed and an appropriate choice of feedback gains and time delay is discussed from the viewpoint of vibration control. In order to numerically solve the problem of history dependence prior to the start of excitation, the concepts of PoincarĂ© section and of fixed points are generalized. Then, a modified shooting scheme associated with the path following technique is proposed to locate periodic motions of the delay system. Numerical results show the efficiency of the first-order approximations to resonances”.
Reviewer: I.Ginchev (Varna)

34K20 Stability theory of functional-differential equations
70J35 Forced motions in linear vibration theory
70J25 Stability for problems in linear vibration theory
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