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Control of Bistability in a Delayed Duffing Oscillator

Control of Bistability in a Delayed Duffing Oscillator Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2012, Article ID 872498, 5 pages doi:10.1155/2012/872498 Research Article Control of Bistability in a Delayed Duffing Oscillator Mustapha Hamdi and Mohamed Belhaq Laboratory of Mechanics, Hassan II University Casablanca, Morocco Correspondence should be addressed to Mohamed Belhaq, mbelhaq@yahoo.fr Received 25 June 2011; Accepted 19 September 2011 Academic Editor: Marek Pawelczyk Copyright © 2012 M. Hamdi and M. Belhaq. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The effect of a high-frequency excitation on nontrivial solutions and bistability in a delayed Duffing oscillator with a delayed displacement feedback is investigated in this paper. We use the technique of direct partition of motion and the multiple scales method to obtain the slow dynamic of the system and its slow flow. The analysis of the slow flow provides approximations of the Hopf and secondary Hopf bifurcation curves. As a result, this study shows that increasing the delay gain, the system undergoes a secondary Hopf bifurcation. Further, it is indicated that as the frequency of the excitation is increased, the Hopf and secondary Hopf bifurcation curves overlap giving birth in the parameter space to small regions of bistability where a stable trivial steady state and a stable limit cycle coexist. Numerical simulations are carried out to validate the analytical finding. 1. Introduction view point, the influence of high-frequency excitation on nontrivial steady state has not been tackled. In [12], for This paper concerns the effect of a high-frequency excitation instance, a detailed and systematic study on the dynamic of a on the nontrivial solutions and on bistability dynamic of a delayed Duffing oscillator was confined to the linear stability Duffing type oscillator with a delayed displacement feedback analysis. in the form In this paper, we explore the bifurcation of a nontrivial 2 3 solution of (1) created in Hopf bifurcation, and we analyze x¨ + ηx˙ + ω x + βx + λx(t − τ) = 0, (1) the influence of a high-frequency excitation on the bifurca- where η is the damping, ω is the natural frequency, β the tion diagram of this nontrivial state. We show analytically nonlinearity, and λ, τ are the amplitude and the time delay, and numerically that as the frequency of the excitation is respectively. The parameters η, β and λ are assumed to be increased, bistability dynamic appears in small regions in small and positive. This equation may serve as the simplest the parameter space causing the response of the system to model for describing the dynamic of various controlled phys- undergo possible jumps between two steady states. ical and engineering systems; see [1–3]. Other works have been devoted to study the dynamic of a Duffing oscillator 2. Trivial Solution under a delayed feedback control [4–6]. In addition, (1)has been considered in [7] as a simple model for a vibration We begin with a brief review of the stability chart of the trivial problem in turning machine for modelling the nonlinear solution x = 0of (1) by considering its linear version generative effect in metal cutting [1] or for exploring the x¨ + ηx˙ + ω x + λx(t − τ) = 0. (2) control of a flexible beam in a simple mode approach [8]. On the other hand, numerous physical applications focusing The stability analysis of the trivial solution is obtained using on bistability dynamic can be found in the literature [9, 10]. the corresponding transcendental characteristic equation In [1, 11–13], attention has been paid to the linear stability 2 2 −τs s + ηs + ω + λe = 0. (3) analysis of the trivial steady state of delayed oscillators of type (1). While the investigation of stability of nontrivial This equation possesses infinitely many finite roots for λ= 0 solutions in (1) has received little attention from analytical and τ = 0. The stability occurs when two dominant roots of / 2 Advances in Acoustics and Vibration (3) are placed on the imaginary axis at the desired resonant 2 frequency, while other roots remain in the stable left half of 1.8 the complex plane. The imaginary characteristic roots are 1.6 s =±iω ,where ω is the resonance frequency and i = −1. c c The subscript c implies the crossing of the root loci on the 1.4 imaginary axis. We substitute s =±iω into (3), separate the 1.2 real and imaginary parts, eliminate the trigonometric terms U � and we solve for the control parameters λ and τ. This yields the stability diagram corresponding to the Hopf bifurcation 0.8 curves in the (λ, τ) parameter plane with the parametric 0.6 representation • S 0.4 2 2 2 2 λ = ω − ω + ηω , c 0.2 1 ηω 02468 10 12 14 τ = arctan +2( − 1)π , =1, 2, 3,... , 2 2 Time delay, τ dc ω ω − ω (4) Figure 1: Hopf bifurcation curves, taken from [8], for  = 1,... ,4, Ω = 100, η = 0.067, and a = 0.02. 1 1 where  corresponds to the th lobe from the left in the stability diagram illustrated in Figure 1.The family of Hopf 1.5 curves  = 1, 2,... are shown for the given parameters η = Ω = 100 0.067 and a = 0.02. In the dashed region bellow the Hopf curve, the trivial solution is stable. Above this region, the trivial equilibrium is unstable. 0.5 3. Nontrivial Solutions R 0 Following [8], (1) can be viewed as the one mode model of a hinged-clamped beam. Assume that the beam is subjected −0.5 to an axial high-frequency excitation of the form aΩ cos Ωt, where a is a nondimensional amplitude of excitation and Ω is the excitation frequency; the quantity aΩ denotes the −1 excitation strength. Applying the standard method of direct partition of motion [8, 14, 15], we can separate the dynamic −1.5 of (1) into a slow dynamic (at the time-scale of free system 0 0.5 1 1.5 2 2.5 oscillations) and the fast motions (at the rate of the fast Time delay, excitation). Since the slow motions, denoted by the variable Figure 2: Amplitude of the periodic oscillation versus the time z, are of primary concern, the equation describing the slow delay as given by (10), β = 1.25, η = 0.067, λ = 1.2, and a = 0.02. dynamic of the oscillator (1)reads 2 3 z¨ + ηz˙ + ω z + βz + λz(t − τ) = 0, (5) where the independent time scales are defined as T = t where the frequency is now depends on the excitation and T = μt. It follows that the derivatives become d/dt = strength aΩ and given by [8] 2 2 2 2 D + μD and d /dt = D +2μD D + μ D where D = 1 2 1 2 2 n ∂ /∂T . We follow, as usual, the classical steps of the multiple ω = 1+ (aΩ) . (6) scale method by substituting (8) into (7), using the notation j j D = ∂ /∂T , equating coefficients of like powers of μ, n n Now the method of of multiple scales [16]isapplied to and eliminating secular terms. The modulation equations of explore the existence of nontrivial steady state. We assume amplitude R and phase θ of the periodic solutions are given that damping, nonlinearity, and delay are small, and scaling at first-order approximation by the system by introducing a small book-keeping parameter μ,(5)can be dR 1 λ recast as =−μ ηR + μ R sin(ωτ),(9a) dt 2 2ω 2 3 ¨ ˙ ( ) z + ω z + μ ηz + βz + λz t − τ = 0. (7) dθ 3β λ R = μ R + μ R cos(ωτ). (9b) A first-order uniform expansion of the solution to (7)is dt 8ω 2ω sought in the form A fixed point in this slow flow corresponds to a periodic z t; μ = z (T , T ) + μz (T , T ) + O μ ,(8) motion in the original system (7). Solving for the fixed points 1 1 2 2 1 2 Feedback gain, λ c Advances in Acoustics and Vibration 3 0.8 2.5 Ω = 0 0.7 0.6 0.5 1.5 III 0.4 0.3 0.2 II 0.5 0.1 0 5 10 15 20 0 2 4 6 8 10 12 14 τ τ Time delay, Time delay, (a) (b) Ω = 150 4.5 3.5 2.5 IV 1.5 0.5 02468 10 12 Time delay,τ (c) Figure 3: Hopf and secondary Hopf bifurcation curves for different values of Ω; β = 1.25, η = 0.067, and a = 0.02. (a) Ω = 0, Hopf and secondary Hopf curves; (b) Ω = 100, region I: stable trivial solution, region II: stable limit cycle, and region III: quasiperiodic solution; (c) Ω = 150, region IV: multistability domain. of dθ/dt = 0in(9a)and (9b) we obtain the amplitude of the The condition for the nontrivial solution (10)tobereal periodic motion (limit cycle) is (4n − 3)π (4n − 1)π τ  , n = 1, 2,.... (11) 2ω 2ω R = 2 − cos ωτ. (10) Substituting ω by its value given by (6), the condition (11) 3β becomes (4n − 3)π (4n − 1)π τ  , n = 1, 2.... 2 2 Figure 2 plots the variation of this amplitude versus the time 4 4 2 1+ π (aΩ) /2 2 1+ π (aΩ) /2 delay τ. As it can be seen from this figure, the bifurcation (12) value of this limit cycle is τ = π/2. By differentiating once (9a), the stability of the nontrivial Instead of employing the system (9a)and (9b), to obtain solution can be discussed using the corresponding charac- the relation between the amplitude of the periodic solution, teristic equation R,and time delay, τ,asdonein[12], we will take advantage 1 λ from this modulation system to determine the region of s + η − sin ωτ s = 0 (13) 2 2ω existence of this periodic motion born by Hopf bifurcation. Feedback gain, λ Feedback gain, λ Feedback gain, λ 4 Advances in Acoustics and Vibration 1.5 0.05 0.04 0.03 0.02 0.5 0.01 −0.01 −0.5 −0.02 −0.03 −1 −0.04 −0.05 −1.5 0 10 20 30 40 50 60 70 80 90 100 0 102030405060708090 100 Time (t) Time (t) (a) (b) −1 −1 −2 −2 −3 −3 −4 −4 −5 0 102030405060708090 100 0 5 10 15 20 25 30 Time (t) Time (t) (c) (d) Figure 4: Time trace of z(t); β = 1.25, a = 0.02, τ = 5, (a) region I, λ = 0.1, stable trivial solution, (b) region II, λ = 0.8, stable limit cycle, (c) region III, λ = 1.2, quasiperiodic solution, and (d) region IV, λ = 2, τ = 2, multistability solutions. and the critical value for tested sign of the nontrivial when crossing from region I to region II. In the region III eigenvalue (antidashed zone), quasiperiodic oscillations resulting from a secondary Hopf bifurcation take place when crossing from ηω λ = . (14) cr region II to region III. It is worthy to notice that a similar sin ωτ equation to (7) was studied numerically, and it was shown This eigenvalue is negative if λ<λ for any time delay cr that as the delay gain is increased, the system undergoes τ and negative or positive on the rest of the (λ, τ) plane. a secondary Hopf bifurcations [12, 13, 17]. Figure 3(c) The bifurcation curves of the nontrivial steady states given indicates that by increasing the frequency Ω, the Hopf and by (12)and (14), corresponding to the secondary Hopf the secondary Hopf bifurcation curves overlap giving birth bifurcation, are illustrated in Figure 3(a) for Ω = 0. The to regions (dashed region IV in Figure 3(c)) on which a Hopf curves of Figure 1 are also plotted in this figure. In stable trivial steady state and a stable limit cycle coexist. Figures 3(b) and 3(c) are shown the Hopf and the secondary To validate the analytical finding, we show in Figure 4 Hopf curves for Ω = 100 and Ω = 150, respectively. Three numerical time traces integration of (7) corresponding to regions can be distinguished in Figure 3(b). The region I the different regions I, II, III, and IV of Figures 3(b) and (dashed zone) located bellow the Hopf curves corresponds 3(c). Figure 4(a) shows that the stable trivial equilibrium to the domain of stability of the trivial steady state z = in region I loses its stability, and a stable periodic solution 0. The region II (white zone) corresponds to the existence is born by Hopf bifurcation as illustrated in Figure 4(b). domain of a stable limit cycle born by Hopf bifurcation Figure 4(c) indicates the existence of a quasiperiodic solution z(t) z(t) z(t) z(t) Advances in Acoustics and Vibration 5 born by a secondary Hopf bifurcation (region III). Finally, it [11] B. D. Hassard, “Counting roots of the characteristic equation for linear delay-differential systems,” Journal of Differential canbeseenfrom Figure 4(d) (corresponding to region IV) Equations, vol. 136, no. 2, pp. 222–235, 1997. that the trivial stable solution (dotted line) coexists with a [12] H. Wang, H. Hu, and Z. Wang, “Global dynamics of a duffing stable large amplitude limit cycle (solid line) indicating that oscillator with delayed displacement feedback,” International multistability can occur in small regions in the parameter Journal of Bifurcation and Chaos in Applied Sciences and plane (λ, τ). Engineering, vol. 14, no. 8, pp. 2753–2775, 2004. [13] X. Xu, H. Y. Hu, and H. L. Wang, “Stability, bifurcation and chaos of a delayed oscillator with negative damping and 4. Conclusion delayed feedback control,” Nonlinear Dynamics, vol. 49, no. 1- 2, pp. 117–129, 2007. We have investigated the effect of a high-frequency excitation [14] I. I. Blekhman, Vibrational Mechanics Nonlinear Dynamic on nontrivial steady-state solutions and bistability in a Effects, General Approach, Application? World Scientific, Sin- delayed Duffing oscillator. The technique of direct partition gapore, 2000. of motion and the multiple scales method were applied [15] J. J. Thomsen, Vibrations and Stability: Advanced Theory, to obtain the equation governing the slow dynamic of the Analysis, and Tools, Springer, Berlin-Heidelberg, Germany, oscillator and the corresponding slow flow. The nontrivial solutions of the slow flow were studied, and the secondary [16] A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley, Hopf bifurcation curves were obtained. It was shown that a New York, NY, USA, 1979. high-frequency excitation causes the Hopf and the secondary [17] T. Erneux and T. Kalmar ´ -Nagy, “Nonlinear stability of a Hopf diagrams to overlap giving rise to small regions in the delayed feedback controlled container crane,” Journal of Vibra- parameter space control (gain versus time delay) where a tion and Control, vol. 13, no. 5, pp. 603–616, 2007. stable equilibrium and a stable large amplitude limit cycle may coexist. This coexistence may produce possible jumps between the two steady states. This bistability regime can be either desirable or undesirable depending on the application under consideration. The analytical result of this work has been confirmed using numerical simulations. References [1] G. Step ´ an, ´ Retarded Dynamical Systems: Sability and Charac- teristic Functions, Longman Scientific and Technical, 1989. [2] A. Nayfeh, C. Chin, and J. Pratt, “Applications of perturbation methods to tool chatter dynamics,” in Dynamics and Chaos in Manufacturing Processes, F. C. Moon, Ed., pp. 193–213, Wiley, [3] H. Y. Hu and Z. H. Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer, 2002. [4] X. Li, J. C. Ji, C. H. Hansen, and C. Tan, “The response of a Duffing-van der Pol oscillator under delayed feedback control,” Journal of Sound and Vibration, vol. 291, no. 3–5, pp. 644–655, 2006. [5] S. Chatterjee, “Self-excited oscillation under nonlinear feed- back with time-delay,” Journal of Sound and Vibration, vol. 330, no. 9, pp. 1860–1876, 2011. [6] K. A. Alhazza, A. H. Nayfeh, and M. F. Daqaq, “On utilizing delayed feedback for active-multimode vibration control of cantilever beams,” Journal of Sound and Vibration, vol. 319, no. 3–5, pp. 735–752, 2009. [7] M.A.Johnson, Nonlinear differential equations with delay as models for vibrations in the machining of metals, Ph.D. thesis, Cornell University, 1996. [8] M. Hamdi and M. Belhaq, “Self-excited vibration control for axially fast excited beam by a time delay state feedback,” Chaos, Solitons and Fractals, vol. 41, no. 2, pp. 521–532, 2009. [9] S.M.Houghton, S. M. Tobias,E.Knobloch, andM.R. E. Proctor, “Bistability in the complex Ginzburg-Landau equation with drift,” Physica D, vol. 238, no. 2, pp. 184–196, [10] P. Jung and P. Hangg ¨ i, “Bistability and colored noise in nonequilibrium systems: theory versus precise numerics,” Physical Review Letters, vol. 61, no. 1, pp. 11–14, 1988. 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Control of Bistability in a Delayed Duffing Oscillator

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Copyright © 2012 Mustapha Hamdi and Mohamed Belhaq. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2012, Article ID 872498, 5 pages doi:10.1155/2012/872498 Research Article Control of Bistability in a Delayed Duffing Oscillator Mustapha Hamdi and Mohamed Belhaq Laboratory of Mechanics, Hassan II University Casablanca, Morocco Correspondence should be addressed to Mohamed Belhaq, mbelhaq@yahoo.fr Received 25 June 2011; Accepted 19 September 2011 Academic Editor: Marek Pawelczyk Copyright © 2012 M. Hamdi and M. Belhaq. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The effect of a high-frequency excitation on nontrivial solutions and bistability in a delayed Duffing oscillator with a delayed displacement feedback is investigated in this paper. We use the technique of direct partition of motion and the multiple scales method to obtain the slow dynamic of the system and its slow flow. The analysis of the slow flow provides approximations of the Hopf and secondary Hopf bifurcation curves. As a result, this study shows that increasing the delay gain, the system undergoes a secondary Hopf bifurcation. Further, it is indicated that as the frequency of the excitation is increased, the Hopf and secondary Hopf bifurcation curves overlap giving birth in the parameter space to small regions of bistability where a stable trivial steady state and a stable limit cycle coexist. Numerical simulations are carried out to validate the analytical finding. 1. Introduction view point, the influence of high-frequency excitation on nontrivial steady state has not been tackled. In [12], for This paper concerns the effect of a high-frequency excitation instance, a detailed and systematic study on the dynamic of a on the nontrivial solutions and on bistability dynamic of a delayed Duffing oscillator was confined to the linear stability Duffing type oscillator with a delayed displacement feedback analysis. in the form In this paper, we explore the bifurcation of a nontrivial 2 3 solution of (1) created in Hopf bifurcation, and we analyze x¨ + ηx˙ + ω x + βx + λx(t − τ) = 0, (1) the influence of a high-frequency excitation on the bifurca- where η is the damping, ω is the natural frequency, β the tion diagram of this nontrivial state. We show analytically nonlinearity, and λ, τ are the amplitude and the time delay, and numerically that as the frequency of the excitation is respectively. The parameters η, β and λ are assumed to be increased, bistability dynamic appears in small regions in small and positive. This equation may serve as the simplest the parameter space causing the response of the system to model for describing the dynamic of various controlled phys- undergo possible jumps between two steady states. ical and engineering systems; see [1–3]. Other works have been devoted to study the dynamic of a Duffing oscillator 2. Trivial Solution under a delayed feedback control [4–6]. In addition, (1)has been considered in [7] as a simple model for a vibration We begin with a brief review of the stability chart of the trivial problem in turning machine for modelling the nonlinear solution x = 0of (1) by considering its linear version generative effect in metal cutting [1] or for exploring the x¨ + ηx˙ + ω x + λx(t − τ) = 0. (2) control of a flexible beam in a simple mode approach [8]. On the other hand, numerous physical applications focusing The stability analysis of the trivial solution is obtained using on bistability dynamic can be found in the literature [9, 10]. the corresponding transcendental characteristic equation In [1, 11–13], attention has been paid to the linear stability 2 2 −τs s + ηs + ω + λe = 0. (3) analysis of the trivial steady state of delayed oscillators of type (1). While the investigation of stability of nontrivial This equation possesses infinitely many finite roots for λ= 0 solutions in (1) has received little attention from analytical and τ = 0. The stability occurs when two dominant roots of / 2 Advances in Acoustics and Vibration (3) are placed on the imaginary axis at the desired resonant 2 frequency, while other roots remain in the stable left half of 1.8 the complex plane. The imaginary characteristic roots are 1.6 s =±iω ,where ω is the resonance frequency and i = −1. c c The subscript c implies the crossing of the root loci on the 1.4 imaginary axis. We substitute s =±iω into (3), separate the 1.2 real and imaginary parts, eliminate the trigonometric terms U � and we solve for the control parameters λ and τ. This yields the stability diagram corresponding to the Hopf bifurcation 0.8 curves in the (λ, τ) parameter plane with the parametric 0.6 representation • S 0.4 2 2 2 2 λ = ω − ω + ηω , c 0.2 1 ηω 02468 10 12 14 τ = arctan +2( − 1)π , =1, 2, 3,... , 2 2 Time delay, τ dc ω ω − ω (4) Figure 1: Hopf bifurcation curves, taken from [8], for  = 1,... ,4, Ω = 100, η = 0.067, and a = 0.02. 1 1 where  corresponds to the th lobe from the left in the stability diagram illustrated in Figure 1.The family of Hopf 1.5 curves  = 1, 2,... are shown for the given parameters η = Ω = 100 0.067 and a = 0.02. In the dashed region bellow the Hopf curve, the trivial solution is stable. Above this region, the trivial equilibrium is unstable. 0.5 3. Nontrivial Solutions R 0 Following [8], (1) can be viewed as the one mode model of a hinged-clamped beam. Assume that the beam is subjected −0.5 to an axial high-frequency excitation of the form aΩ cos Ωt, where a is a nondimensional amplitude of excitation and Ω is the excitation frequency; the quantity aΩ denotes the −1 excitation strength. Applying the standard method of direct partition of motion [8, 14, 15], we can separate the dynamic −1.5 of (1) into a slow dynamic (at the time-scale of free system 0 0.5 1 1.5 2 2.5 oscillations) and the fast motions (at the rate of the fast Time delay, excitation). Since the slow motions, denoted by the variable Figure 2: Amplitude of the periodic oscillation versus the time z, are of primary concern, the equation describing the slow delay as given by (10), β = 1.25, η = 0.067, λ = 1.2, and a = 0.02. dynamic of the oscillator (1)reads 2 3 z¨ + ηz˙ + ω z + βz + λz(t − τ) = 0, (5) where the independent time scales are defined as T = t where the frequency is now depends on the excitation and T = μt. It follows that the derivatives become d/dt = strength aΩ and given by [8] 2 2 2 2 D + μD and d /dt = D +2μD D + μ D where D = 1 2 1 2 2 n ∂ /∂T . We follow, as usual, the classical steps of the multiple ω = 1+ (aΩ) . (6) scale method by substituting (8) into (7), using the notation j j D = ∂ /∂T , equating coefficients of like powers of μ, n n Now the method of of multiple scales [16]isapplied to and eliminating secular terms. The modulation equations of explore the existence of nontrivial steady state. We assume amplitude R and phase θ of the periodic solutions are given that damping, nonlinearity, and delay are small, and scaling at first-order approximation by the system by introducing a small book-keeping parameter μ,(5)can be dR 1 λ recast as =−μ ηR + μ R sin(ωτ),(9a) dt 2 2ω 2 3 ¨ ˙ ( ) z + ω z + μ ηz + βz + λz t − τ = 0. (7) dθ 3β λ R = μ R + μ R cos(ωτ). (9b) A first-order uniform expansion of the solution to (7)is dt 8ω 2ω sought in the form A fixed point in this slow flow corresponds to a periodic z t; μ = z (T , T ) + μz (T , T ) + O μ ,(8) motion in the original system (7). Solving for the fixed points 1 1 2 2 1 2 Feedback gain, λ c Advances in Acoustics and Vibration 3 0.8 2.5 Ω = 0 0.7 0.6 0.5 1.5 III 0.4 0.3 0.2 II 0.5 0.1 0 5 10 15 20 0 2 4 6 8 10 12 14 τ τ Time delay, Time delay, (a) (b) Ω = 150 4.5 3.5 2.5 IV 1.5 0.5 02468 10 12 Time delay,τ (c) Figure 3: Hopf and secondary Hopf bifurcation curves for different values of Ω; β = 1.25, η = 0.067, and a = 0.02. (a) Ω = 0, Hopf and secondary Hopf curves; (b) Ω = 100, region I: stable trivial solution, region II: stable limit cycle, and region III: quasiperiodic solution; (c) Ω = 150, region IV: multistability domain. of dθ/dt = 0in(9a)and (9b) we obtain the amplitude of the The condition for the nontrivial solution (10)tobereal periodic motion (limit cycle) is (4n − 3)π (4n − 1)π τ  , n = 1, 2,.... (11) 2ω 2ω R = 2 − cos ωτ. (10) Substituting ω by its value given by (6), the condition (11) 3β becomes (4n − 3)π (4n − 1)π τ  , n = 1, 2.... 2 2 Figure 2 plots the variation of this amplitude versus the time 4 4 2 1+ π (aΩ) /2 2 1+ π (aΩ) /2 delay τ. As it can be seen from this figure, the bifurcation (12) value of this limit cycle is τ = π/2. By differentiating once (9a), the stability of the nontrivial Instead of employing the system (9a)and (9b), to obtain solution can be discussed using the corresponding charac- the relation between the amplitude of the periodic solution, teristic equation R,and time delay, τ,asdonein[12], we will take advantage 1 λ from this modulation system to determine the region of s + η − sin ωτ s = 0 (13) 2 2ω existence of this periodic motion born by Hopf bifurcation. Feedback gain, λ Feedback gain, λ Feedback gain, λ 4 Advances in Acoustics and Vibration 1.5 0.05 0.04 0.03 0.02 0.5 0.01 −0.01 −0.5 −0.02 −0.03 −1 −0.04 −0.05 −1.5 0 10 20 30 40 50 60 70 80 90 100 0 102030405060708090 100 Time (t) Time (t) (a) (b) −1 −1 −2 −2 −3 −3 −4 −4 −5 0 102030405060708090 100 0 5 10 15 20 25 30 Time (t) Time (t) (c) (d) Figure 4: Time trace of z(t); β = 1.25, a = 0.02, τ = 5, (a) region I, λ = 0.1, stable trivial solution, (b) region II, λ = 0.8, stable limit cycle, (c) region III, λ = 1.2, quasiperiodic solution, and (d) region IV, λ = 2, τ = 2, multistability solutions. and the critical value for tested sign of the nontrivial when crossing from region I to region II. In the region III eigenvalue (antidashed zone), quasiperiodic oscillations resulting from a secondary Hopf bifurcation take place when crossing from ηω λ = . (14) cr region II to region III. It is worthy to notice that a similar sin ωτ equation to (7) was studied numerically, and it was shown This eigenvalue is negative if λ<λ for any time delay cr that as the delay gain is increased, the system undergoes τ and negative or positive on the rest of the (λ, τ) plane. a secondary Hopf bifurcations [12, 13, 17]. Figure 3(c) The bifurcation curves of the nontrivial steady states given indicates that by increasing the frequency Ω, the Hopf and by (12)and (14), corresponding to the secondary Hopf the secondary Hopf bifurcation curves overlap giving birth bifurcation, are illustrated in Figure 3(a) for Ω = 0. The to regions (dashed region IV in Figure 3(c)) on which a Hopf curves of Figure 1 are also plotted in this figure. In stable trivial steady state and a stable limit cycle coexist. Figures 3(b) and 3(c) are shown the Hopf and the secondary To validate the analytical finding, we show in Figure 4 Hopf curves for Ω = 100 and Ω = 150, respectively. Three numerical time traces integration of (7) corresponding to regions can be distinguished in Figure 3(b). The region I the different regions I, II, III, and IV of Figures 3(b) and (dashed zone) located bellow the Hopf curves corresponds 3(c). Figure 4(a) shows that the stable trivial equilibrium to the domain of stability of the trivial steady state z = in region I loses its stability, and a stable periodic solution 0. The region II (white zone) corresponds to the existence is born by Hopf bifurcation as illustrated in Figure 4(b). domain of a stable limit cycle born by Hopf bifurcation Figure 4(c) indicates the existence of a quasiperiodic solution z(t) z(t) z(t) z(t) Advances in Acoustics and Vibration 5 born by a secondary Hopf bifurcation (region III). Finally, it [11] B. D. Hassard, “Counting roots of the characteristic equation for linear delay-differential systems,” Journal of Differential canbeseenfrom Figure 4(d) (corresponding to region IV) Equations, vol. 136, no. 2, pp. 222–235, 1997. that the trivial stable solution (dotted line) coexists with a [12] H. Wang, H. Hu, and Z. Wang, “Global dynamics of a duffing stable large amplitude limit cycle (solid line) indicating that oscillator with delayed displacement feedback,” International multistability can occur in small regions in the parameter Journal of Bifurcation and Chaos in Applied Sciences and plane (λ, τ). Engineering, vol. 14, no. 8, pp. 2753–2775, 2004. [13] X. Xu, H. Y. Hu, and H. L. Wang, “Stability, bifurcation and chaos of a delayed oscillator with negative damping and 4. Conclusion delayed feedback control,” Nonlinear Dynamics, vol. 49, no. 1- 2, pp. 117–129, 2007. We have investigated the effect of a high-frequency excitation [14] I. I. Blekhman, Vibrational Mechanics Nonlinear Dynamic on nontrivial steady-state solutions and bistability in a Effects, General Approach, Application? World Scientific, Sin- delayed Duffing oscillator. The technique of direct partition gapore, 2000. of motion and the multiple scales method were applied [15] J. J. Thomsen, Vibrations and Stability: Advanced Theory, to obtain the equation governing the slow dynamic of the Analysis, and Tools, Springer, Berlin-Heidelberg, Germany, oscillator and the corresponding slow flow. The nontrivial solutions of the slow flow were studied, and the secondary [16] A. H. Nayfeh and D. T. Mook, Nonlinear Oscillations, Wiley, Hopf bifurcation curves were obtained. It was shown that a New York, NY, USA, 1979. high-frequency excitation causes the Hopf and the secondary [17] T. Erneux and T. Kalmar ´ -Nagy, “Nonlinear stability of a Hopf diagrams to overlap giving rise to small regions in the delayed feedback controlled container crane,” Journal of Vibra- parameter space control (gain versus time delay) where a tion and Control, vol. 13, no. 5, pp. 603–616, 2007. stable equilibrium and a stable large amplitude limit cycle may coexist. This coexistence may produce possible jumps between the two steady states. This bistability regime can be either desirable or undesirable depending on the application under consideration. The analytical result of this work has been confirmed using numerical simulations. References [1] G. Step ´ an, ´ Retarded Dynamical Systems: Sability and Charac- teristic Functions, Longman Scientific and Technical, 1989. [2] A. Nayfeh, C. Chin, and J. Pratt, “Applications of perturbation methods to tool chatter dynamics,” in Dynamics and Chaos in Manufacturing Processes, F. C. Moon, Ed., pp. 193–213, Wiley, [3] H. Y. Hu and Z. H. Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer, 2002. [4] X. Li, J. C. Ji, C. H. Hansen, and C. Tan, “The response of a Duffing-van der Pol oscillator under delayed feedback control,” Journal of Sound and Vibration, vol. 291, no. 3–5, pp. 644–655, 2006. [5] S. Chatterjee, “Self-excited oscillation under nonlinear feed- back with time-delay,” Journal of Sound and Vibration, vol. 330, no. 9, pp. 1860–1876, 2011. [6] K. A. Alhazza, A. H. Nayfeh, and M. F. Daqaq, “On utilizing delayed feedback for active-multimode vibration control of cantilever beams,” Journal of Sound and Vibration, vol. 319, no. 3–5, pp. 735–752, 2009. [7] M.A.Johnson, Nonlinear differential equations with delay as models for vibrations in the machining of metals, Ph.D. thesis, Cornell University, 1996. [8] M. Hamdi and M. Belhaq, “Self-excited vibration control for axially fast excited beam by a time delay state feedback,” Chaos, Solitons and Fractals, vol. 41, no. 2, pp. 521–532, 2009. [9] S.M.Houghton, S. M. Tobias,E.Knobloch, andM.R. E. Proctor, “Bistability in the complex Ginzburg-Landau equation with drift,” Physica D, vol. 238, no. 2, pp. 184–196, [10] P. Jung and P. Hangg ¨ i, “Bistability and colored noise in nonequilibrium systems: theory versus precise numerics,” Physical Review Letters, vol. 61, no. 1, pp. 11–14, 1988. 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