Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems with time delay

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(1)Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems with time delay Mustapha Hamdi & Mohamed Belhaq. Nonlinear Dynamics An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems ISSN 0924-090X Nonlinear Dyn DOI 10.1007/s11071-013-0762-6. 1 23.

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(3) Author's personal copy Nonlinear Dyn DOI 10.1007/s11071-013-0762-6. O R I G I N A L PA P E R. Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems with time delay Mustapha Hamdi · Mohamed Belhaq. Received: 15 September 2012 / Accepted: 4 January 2013 © Springer Science+Business Media Dordrecht 2013. Abstract The effect of time-delayed feedback and fast harmonic excitation (FHE) on stationary periodic vibration and quasi-periodic responses in a parametric and self-excited weakly nonlinear oscillator is analyzed in this paper. The method of direct partition of motion and two stages of multiple scales analysis are conducted to obtain analytical approximation for quasi-periodic oscillation envelopes and frequencylocking area near primary resonance. A parameter study shows that, in the absence or the presence of high-frequency excitation, time-delayed feedback may reduce significantly the amplitude and the envelopes of quasi-periodic oscillations leading to a quasi synchronization of the response over the whole frequency range around the resonance. The results presented for the parameters tested agree well with results obtained by numerical simulation. Keywords Self-excitation · Parametric excitation · Time delay · Frequency-locking · Quasi-periodic vibration · Control 1 Introduction In self-excited mechanical systems driven by periodic forcing the frequency-locking phenomenon (or synM. Hamdi · M. Belhaq () Laboratory of Mechanics, University Hassan II-Aïn Chock, PB 5366, Maârif, Casablanca, Morocco e-mail: [email protected]. chronization) can occur in a small region near resonances leading to frequency-locked oscillations for which the system vibrates with the forcing frequency; see for instance [1–13] and references herein. Away from the resonance the amplitude of vibrations is modulated with a certain frequency such that the response of the system is quasi-periodic (QP). When approaching the resonance region by sweeping the forcing frequency backward or forward, the modulation of the amplitude disappears abruptly via a jump giving rise to frequency-locked oscillations corresponding to the stable branch of the frequency-response curve. This jump in the system response from QP to periodic vibrations or vice versa is associated with the coexistence of two stable stationary solutions in the slow flow (equilibrium and limit cycle) and its control is of importance in applications involving parametric and self-excited vibrations in which frequency-locking is desired. A specific example of application that may produce such a phenomenon is the regenerative effect in high-speed milling. High speed milling can induce a parametric excitation and milling itself can generate self-oscillations [14]. Recently, the problem of frequency-locking and jumps in self-excited and forced oscillators has received more attention. In [15], the control of jumps in a forced van der Pol–Duffing oscillator was studied near the primary resonance and it was shown that FHE can suppress jumps in a certain range of the frequency. Similar studies have been done near primary and secondary resonances for a Mathieu–van der.

(4) Author's personal copy M. Hamdi, M. Belhaq. Pol–Duffing (MvdPD) oscillator [16, 17] and it was shown that FHE changes the nonlinear characteristic of the system from softening to hardening and shifts the frequency-locked area toward lower frequencies. On the other hand, numerous studies on the influence of delay on self-excited nonlinear systems have been carried out; see for instance [18–21]. Note also that a delayed feedback was used to quench undesirable vibrations in a van der Pol type system [22–25]. To the best of our knowledge, only few works examined the interaction effect of time delay and FHE on the dynamic of self-excited nonlinear systems. Specifically, such an interaction effect was analyzed in [22, 23, 25] taking into account the tilting effect of the fast excitation [25]. It was shown that time delay, and the incline of the FHE can be exploited separately or simultaneously to control the self-excited vibrations. This work is motivated by the important issue of capturing frequency-locking in parametrically and self-excited mechanical systems evolving in a highfrequency vibrational environment that may be induced by some external vibrational sources. Such a phenomenon may occur in many mechanical systems, such as manufacturing processes in high-speed milling, flow-induced vibrations and vibrations in rotor systems. The aim of the present paper is to approximate analytically the QP modulation envelop and frequencylocked area for a fast excited MvdPD oscillator near primary resonance. The study can be viewed as an extension of results given in [16] in which time delay was not considered. The analytical approach to be developed here is the so-called three stages perturbation analysis (TSPA). This approach consists of applying successively the method of direct partition of motion (DPM) [29] and two stages of multiple scale method (MSM) [26, 27] required to capture the envelop of the QP vibrations. In the next section, the method of DPM is applied to drive the equation governing the slow dynamic. Note that this averaging method is equivalent to MSM [26, 27] and thereby to the method of Kuzmak [28]. In Sect. 3, a first MSM is performed on the slow dynamic to obtain the corresponding slow flow system. The bifurcation curves of periodic solutions and the frequency response are also given. In Sect. 4, a second MSM is implemented on the slow flow to approximate the slow flow limit cycle and the QP oscillation envelop. The effect of time delay on QP modulation. envelop and on frequency-locked area is also analyzed in this section. Section 5 concludes the work.. 2 Equation of motion and slow dynamic Consider the MvdPD oscillator subjected to a horizontal FHE and cubic time delay in the form ẍ + (1 − h cos ωt)x − α − βx 2 ẋ − γ x 3 + λ1 x(t − τd ) − λ2 x 3 (t − τd ) = aΩ 2 cos x cos Ωt. (1). where α, β are damping coefficients, γ is the nonlinearity component, h and a are the excitation amplitudes and λ1 and λ2 are the linear and the nonlinear feedback gains, respectively, while τd is the time delay. An overdot denotes differentiation with respect to time t and it is assumed that the frequency Ω is large comparing to ω such that resonance phenomena between the two frequencies cannot occur. In Eq. (1) different excitations and dynamics are added to a simple harmonic oscillator. Concerning the excitations, Eq. (1) combines a self-excited excitation modeled by a van der Pol term, −(α − βx 2 )ẋ [26], a parametric forcing given by the Mathieu term, −h cos (ωt)x [26], and a FHE modeled by aΩ 2 cos x cos Ωt [29]. The latter term could originate, e.g., for a pendulum with a point of support horizontally shaken at a high-frequency and with large displacement amplitude. Note that the general case of a titled excitation including the particular limiting cases of horizontal and vertical excitations was considered in [25]. In addition to the forcing terms, Eq. (1) includes a cubic restoring force and linear plus nonlinear time-delayed feedback modeling the dependence of temporal evolution on information from the past [30]. This time delay arises, for instance, in active feedback control systems and indicates the period of time between the instant the feedback signal is measured and actual system actuation [31, 32]. There are numerous examples that can be modeled by Eq. (1) like, modulation of flutter of aircraft wings, friction-induced vibrations in modulated brakes and clutches, chattering of modulated machine tools, etc. Time delay components can be added as an active control device for suppressing or avoiding some undesirable oscillations. Note that a weakly nonlinear delay systems with cubic nonlinearities has been considered for analyzing the dynamic of the system with arbitrarily large gains [33]..

(5) Author's personal copy Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems. To derive the slow dynamic of system (1), we use the method of DPM [29] by introducing two different time scales, a fast time T0 = Ωt and a slow time T1 = t, and we split up x(t) as. An approximate expression for φ is obtained from (5) by considering only the dominant terms of order −1 as. x(t) = z(T1 ) + φ(T0 , T1 ). where it is assumed that aΩ = O( The stationary solution to the first order for φ is written as. (2). where z describes the slow main motion at time-scale of oscillations and φ stands for an overlay of the fast motions where indicates that φ is small compared to z. Since Ω is assumed to be a large parameter, we choose ≡ Ω −1 and we suppose that the fast part φ and its derivatives are 2π -periodic functions of fast time T0 with zero mean value with respect 2πto this time, 1 so that x(t) = z(T1 ) where ≡ 2π 0 () dT0 defines time-averaging operator over one period of the fast excitation with the slow time T1 fixed. j j d Introducing Di ≡ ∂∂j T yields dt = −1 D0 + D1 , d2 dt 2. = −2 D02 + 2 into (1) gives −1. −1 D. i. 0 D1. + D12 and substituting (2). D02 φ + 2D0 D1 φ + D12 φ + D12 z + (1 − h cos ωT1 )(z + φ) − α − β(z + φ)2 (D0 φ + D1 φ + D1 z) − γ (z + φ)3 + λ1 z(T1 − τd ) + φ 3 − λ2 z(T1 − τd ) + φ. =. −1. (aΩ) cos(z + φ) cos T0. (3). Averaging (3) leads to D12 z + (1 − h cos ωT1 )z − α − βz2 D1 z − γ z3 + 3 2 zφ 2 + λ1 z(T1 − τd ) − λ2 z3 (T1 − τd ) = −1 (aΩ) cos(z + φ) cos T0. (aΩ) cos(z + φ) cos T0 − (aΩ) cos(z + φ) cos T0. (7). The equation governing the slow motion is derived from (4). Inserting cos(z + φ) = cos z − φ sin z + O( 2 ) into Eq. (4) and retaining the dominant terms of order 0 , we obtain D12 z + (1 − h cos ωT1 )z − α − βz2 D1 z − γ z3 + λ1 z(T1 − τd ) − λ2 z3 (T1 − τd ) = −(aΩ) sin zφ cos T0 . (8). from (7) and using that cos2 T. Inserting φ 0 = 1/2 we find the equation for slow motions D12 z + (1 − h cos ωT1 )z − α − βz2 D1 z − γ z3 + λ1 z(T1 − τd ) − λ2 z3 (T1 − τd ) 1 = (aΩ)2 cos z sin z (9) 2 Considering small vibrations around the origin by expanding in Taylor’s series the terms sin z and cos z and keeping only terms up to order three in z, Eq. (9) becomes D12 z + ω02 − h cos ωT1 z 1 2 2 − α − βz D1 z − γ − (aΩ) z3 3 (10). (4) 3 Slow flow and frequency response Equation (10) can be rewritten in the form z̈ + ω02 z = α − βz2 ż + ξ z3 − λ1 z(t − τd ) + λ2 z3 (t − τd ) + hz cos ωt. −1. −1. φ = −a cos z cos T0. where ω02 = 1 − 12 (aΩ)2 .. D02 φ + 2D0 D1 φ + D12 φ + (1 − h cos ωT1 ) φ − α − βz2 (D0 φ + D1 φ) + β 2 φ + 2 φ 2 (D0 φ + D1 φ) − γ 3 z 2 φ + 3 φ 3 + λ1 φ − λ2 3 z2 (T1 − τd )φ + 3 2 z(T1 − τd )φ 2 + 3 φ 3. =. (6) 0 ).. + λ1 z(T1 − τd ) − λ2 z3 (T1 − τd ) = 0. Subtracting (4) from (3) yields −1. D02 φ = (aΩ) cos z cos T0. (5). (11). where ξ = γ − 13 (aΩ)2 and an overdot denotes differentiation with respect to time t. We express the primary resonance condition by introducing a detuning parameter σ according to 2 ω ω02 = +σ (12) 2.

(6) Author's personal copy M. Hamdi, M. Belhaq. Next we apply the MSM [26] to drive the equation of modulation near the primary resonance. This can be done by introducing a bookkeeping small parameter μ and rewriting Eq. (11) in the form z̈ +. ω2 4. z = μ −σ z + α − βz2 ż + ξ z3 − λ1 z(T1 − τd ) + λ2 z3 (T1 − τd ) + hz cos ωT1. (13). A solution to Eq. (13) can be sought in the form (14) z(t) = z0 (T1 , T2 ) + μz1 (T1 , T2 ) + O μ2 where T1 = t as defined before and T2 = μt. In terms d of the variables Ti , the time derivatives become dt = D1 + μD2 + O(μ2 ) and ∂j. j. d2 dt 2. = D12 + 2μD1 D2 +. O(μ2 ) where Di = ∂ j T . Substituting Eq. (14) into i Eq. (13) and equating coefficients of like powers of μ, we obtain at different orders of μ D12 z0 +. ω2 z0 = 0 4. (15). ω2 z1 4 = −2D1 D2 z0 − σ z0 + α − βz02 D1 z0 + ξ z03. D12 z1 +. − λ1 z0 (T1 − τd ) + λ2 z03 (T1. − τd ). + hz0 cos ωT1. (16). The solution to the first order is given by iω z0 (T1 , T2 ) = A(T2 ) exp T1 2 iω + Ā(T2 ) exp − T1 2. (17). where A and Ā are complex conjugate functions. This solution is substituted into Eq. (16) to obtain ω2 z1 D12 z1 + 4. −iωD2 A − σ A + i. βω 2 αω A−i A Ā + 3ξ A2 Ā 2 2. iωτd iωτd h (19) + Ā − λ1 Ae− 2 + 3λ2 A2 Āe− 2 = 0 2 The function A may be expressed in the polar form. 1 (20) A = R exp(iθ ) 2 where R and θ are real functions with respect to T2 . Substituting the expression of A into (19) and separating the real and imaginary parts, we obtain the slow flow system h dR R sin(2θ ) = k1 R + k2 R 3 − dT2 2ω dθ h cos(2θ ) = k3 + k4 R 2 − dT2 2ω. (21). ωτd 2 where k1 = α2 + λω1 sin ωτ2 d , k2 = − β8 − 3λ 4ω sin 2 , 3ξ ωτd 2 − 3λ k3 = ωσ + λω1 cos ωτ2 d and k4 = − 4ω 4ω cos 2 . Since the slow flow (21) is invariant under the transformation θ → −θ + π2 , k3 → −k3 , and k4 → −k4 , then the slow flow (21) can be written in the form. h dR R sin(2θ ) = k1 R + k2 R 3 − dT2 2ω dθ h cos(2θ ) = sk3 + sk4 R 2 − dT2 2ω. (22). where s = ±1. This invariance property will be used to approximate the envelope of the modulation amplitude of the QP vibrations. The fixed points of the system (22), corresponding to periodic oscillations of dR dθ = dT = 0. Eq. (11), are determined by setting dT 2 2 2 Eliminating θ and define ρ = R , we obtain the following quadratic equation on ρ: Aρ 2 − 2Bρ + C = 0. (23). where A = k22 + k42 , B = −k1 k2 − k3 k4 and C = k12 + 2. βω 2 αω A−i A Ā = −iωD2 A − σ A + i 2 2 iωτd h + 3ξ A2 Ā + Ā − λ1 Ae− 2 2 + 3λ2 A2 Āe−. and NST represent parts that do not produce secular terms. Eliminating the terms that produce secular terms from (18), we have. iωτd 2. e. iω 2 T1. + cc + NST. (18). where symbol cc denotes the parts of the complex conjugate of the function at right-hand side of Eq. (18),. h k32 − 4ω 2 . The real solution ρ of Eq. (23) determines the amplitude-frequency response of the slow dynamic (10). Equation (23) has two real roots if the discriminant Δ is non negative. These two solutions are positive if the conditions C > 0 and B > 0 are held. On the other hand, Eq. (23) has only one positive root if the condition C < 0 is satisfied. Next we fix the parameters α = 0.01, β = 0.05, γ = 0.1, h = 0.1 and a = 0.01..

(7) Author's personal copy Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems. Fig. 1 Bifurcation curves of periodic solutions of the slow dynamic (11) near primary resonance in the absence of FHE; Ω = 0 (a) λ1 = λ2 = τd = 0, (b) λ1 = 0.1, λ2 = 0.5, τd = 1.5. Fig. 2 Bifurcation curves of periodic solutions of the slow dynamic (11) near primary resonance; parameter values as in Fig. 1, except that Ω = 100. Figure 1 shows, in the (h, ω)-plane, the bifurcation curves of periodic solutions in the absence of FHE (Ω = 0) for the undelayed (Fig. 1a) and the delayed (Fig. 1b) cases. Figure 2 illustrates similar curves but for Ω = 100. In these figures, three regions can be distinguished. In region I, where the conditions Δ < 0 and C < 0 are satisfied, there are two possible solutions: an unstable trivial solution and stable non-trivial solution, while within region II, where the conditions Δ < 0, B > 0 and C > 0 are satisfied, there are three possible solutions: the unstable trivial solution, stable and unstable non-trivial solutions. Within regions III. and IV an unstable trivial solution and a stable limit cycle exist. We note that Figs. 1 and 2 give a global view on the effect of the FHE and time delay on the bifurcation curves when they are acted separately or simultaneously. For instance, in Fig. 1 can be seen the influence of time delay in the absence of FHE, in Fig. 1a and Fig. 2a the effect of FHE in the absence of time delay (as shown in [16]) and in Fig. 1a and Fig. 2b the simultaneously effect of FHE and time delay. An exchange can be seen from these figures between regions II and III inside the curve Δ = 0 due.

(8) Author's personal copy M. Hamdi, M. Belhaq. Fig. 3 Amplitude response for Ω = 100, ω = 1.43, τd = π (a) vs. λ1 for λ2 = 0.5, (b) vs. λ2 for λ1 = 0.1. Analytical approximation: solid (for stable) and dot lines (for instable). Numerical simulation: circles. Fig. 4 Examples of time histories of the slow dynamic z(t) by numerical integration; Ω = 100, ω = 1.43, τd = π (a) λ1 = 0.05 and λ2 = 0.5, (b) λ1 = 0.1 and λ2 = 0.2. to the FHE (Figs. 1a, 2a) and the apparition of a new region IV caused by the delay (Figs. 1b, 2b). Figure 3 shows the variation of the amplitude of the response R versus the linear (Fig. 3a) and nonlinear (Fig. 3b) gains. This requires Eqs. (17) and (20) to be substituted into Eq. (14). Analytical approximations (solid (dot) lines for stable (unstable)) are compared to results obtained by numerical simulations (circles) using Runge–Kutta method. In Fig. 4 we shows time histories of the slow dynamic z(t) obtained by numerical simulation of Eq. (10) corresponding to some values of λ1 and λ2. picked from Fig. 3. These time histories confirm the analytical results given in Fig. 3. Figure 5a shows the variation of the amplitude as function of λ1 in the absence of nonlinear gain (λ2 = 0) and for Ω = 0 and Ω = 100, while Figs. 5b, c illustrate the periodic variation of the amplitude versus time delay for different values of Ω and λ1 , λ2 . Figure 5a indicates that as the frequency Ω increases, the amplitude of the resonance frequency decreases significantly and the behavior of the response changes from softening to hardening. Figures 5b, c show also.

(9) Author's personal copy Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems. Fig. 5 Amplitude response; (a) vs. λ1 for λ2 = 0 and τd = π , (b) vs. τd for λ1 = 0.01 and λ2 = 0, (c) vs. τd for λ1 = 0.01 and λ2 = 0.5. that as Ω increases, the modulation amplitude of the periodic response decreases. Figure 6a illustrates the frequency-response curves for the primary resonance in the absence of time delay (Fig. 6a) and for different values of Ω (as given in [16]). Figure 6b shows the effect of time delay on the frequency-response curves in the presence of the linear gain λ1 (λ2 = 0). Analytical approximations (solid lines) are compared to results obtained by numerical simulations (circles) using Runge–Kutta method. It can be seen that the time delay with small values of linear feedback gain increases the amplitude of the frequency response. In Fig. 7 we plot for different values of the frequency Ω the effects of feedback gains λ1 , λ2 and time delay τd on the frequency-response curves.. Figure 7a, b shows that increasing λ2 decreases significantly the amplitude of the response and inhibits the hardening effect introduced originally by the FHE [16], while Figs. 7b, c, d show that time-delayed feedback introduces a softening effect even in the presence of FHE.. 4 Limit cycle and frequency-locking area In this section we shall approximate the slow flow limit cycle, the QP vibrations envelop and the frequency-locking area. To perform such approximations, it is convenient to transform the polar form (22) to the following Cartesian system, using the variable change u = R cos θ, v = −R sin θ ,.

(10) Author's personal copy M. Hamdi, M. Belhaq. Fig. 6 Effect of the linear gain λ1 on the frequency-response curves for different values of Ω; (a) λ1 = 0, λ2 = 0, τd = 0. Analytical approximation: solid (for stable) and dashed for (instable). Numerical simulation: circles. (b) λ1 = 0.03, λ2 = 0, τd = π/4. du h v = sk3 + dT1 2ω + η k1 u + (k2 u + sk4 v) u2 + v 2 dv h u = − sk3 − dT1 2ω + η k1 v + (k2 v − sk3 u) u2 + v 2. + (k2 v0 − sk4 u0 ) u20 + v02 (24) . where η is a new bookkeeping parameter. To implement the second perturbation analysis, η is introduced in damping and nonlinearity so that the unperturbed system of Eq. (24) admits a basic solution; see Eq. (26). Following [34, 35], we use the MSM to approximate the periodic solution of the slow flow (24). This can be expanded as u(T1 , T2 ) = u0 (T1 , T2 ) + ηu1 (T1 , T2 ) + O η2 (25) v(T1 , T2 ) = v0 (T1 , T2 ) + ηv1 (T1 , T2 ) + O η2 where T1 = t and T2 = ηt. Introducing Di = ∂T∂ i d yields dt = D1 + ηD2 + O(η2 ) and substituting Eqs. (25) into Eqs. (24) and collecting terms, we get the following hierarchy of systems: D12 u0 + ν 2 u0 = 0, h sk3 + v0 = D1 u0 2ω D12 u1 + ν 2 u1 h −D2 v0 + k1 v0 = sk3 + 2ω. − D1 D2 u0 + D2 k1 u0 + (k2 u0 + sk4 v0 ) u20 + v02. (26). (27). h sk3 + v1 2ω = D1 u1 + D2 u0 − k1 u0 − (k2 u0 + sk4 v0 ) u20 + v02. (28). h2. where ν 2 = k32 − 4ω2 is the natural frequency of the slow flow (24) corresponding to the frequency of the periodic solution (slow flow limit cycle). The solution to the first order is given by u0 (T1 , T2 ) = R1 (T2 ) cos νT1 + ϕ(T2 ) v0 (T1 , T2 ) =−. ν. (sk3 +. h 2ω ). (29). R1 (T2 ) sin νT1 + ϕ(T2 ). where R1 and ϕ are, respectively, the amplitude and the phase of the limit cycle. Substituting (29) into (27) and removing secular terms gives the following autonomous slow slow flow system on R1 and ϕ: dR1 2sωk2 k3 3 R = k 1 R1 + dT2 2sωk3 + h 1 sk4 (8ω2 k32 + h2 ) dϕ = R2 dT2 (4sωk + 2h) 8ω2 k 2 − h2 1 3. (30). 3. Equilibria in Eqs. (30) are obtained by setting and given by. dR1 dT2. =0.

(11) Author's personal copy Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems. Fig. 7 Amplitude frequency response near primary resonance; λ1 = 0.01 (a) λ2 = 0.05, τd = π/4, (b) λ2 = 0.5, τd = π/4, (c) λ2 = 0.5, τd = π/2 and (d) λ2 = 0.5, τd = π. R1 = 0,. R1 =. −. k1 (2sωk3 + h) 2sωk2 k3. (31). The non-trivial equilibrium in Eq. (31) corresponds to the amplitude of the slow flow limit cycle and consequently to the modulation amplitude of the QP vibrations in the slow dynamic (11). In Figs. 8a, b we draw for the two different values Ω = 0 and Ω = 100 and in the absence of delay (λ1 = λ2 = τd = 0) the envelopes of the QP vibrations R1 of the slow flow limit cycle, Eq. (31), and the frequency response, as given by Eq. (23). The curves labeled M+ correspond to s = +1 and the curves labeled M− correspond to s = −1 in Eq. (31). The envelopes of QP oscillations obtained by numerical simulations are marked with double circles connected with a vertical line. The com-. parison between the analytical and the numerical approximations shows good agreement, specially away from the resonance. Figure 9 presents examples of time histories of the slow dynamic z(t) obtained by numerical simulation of system (11) for some values of ω picked from Fig. 8a. The time histories show the QP modulation in the response for the two values ω = 1.75 and ω = 2.3. In Figs. 10a, b we plot the envelopes of the QP oscillations and the frequency response for Ω = 0 and Ω = 100, respectively, in the presence of the linear feedback gain λ1 = 0.01 (λ2 = 0) and for τd = π/4. A zoom of the synchronized area of Figs. 10a, b is illustrated in Figs. 11a, b, respectively. These.

(12) Author's personal copy M. Hamdi, M. Belhaq. Fig. 8 Envelope of the modulation amplitude vibration of QP solution and frequency response near the primary resonance; λ1 = λ2 = τd = 0. (a) Ω = 0 and (b) Ω = 100. Fig. 9 Examples of time histories of the slow dynamic z(t) by numerical integration for parameter values as in Fig. 8a except that (a) ω = 1.75, (b) ω = 2.3. figures show that the linear feedback gain λ1 reduces the synchronization area (Figs. 11a, b) and the width of the QP oscillations envelopes as well (Figs. 10a, b). In Figs. 12a, b we plot the same figures as Figs. 10a, b but in the presence of linear and nonlinear feedback gains, λ1 = 0.01, λ2 = 0.8 and for τd = π/4. A zoom of the frequency-locked area of Figs. 12a, b is shown in Figs. 13a, b, respectively. These figures reveal that the nonlinear feedback gain λ2 reduces significantly the amplitude of the resonance re-. sponse and of the width of the QP vibration envelopes (Figs. 12a, b). It can also be seen that the synchronization area is reduced to a very small region presented by a circle at the resonance curve, as shown in Figs. 13a, b. In Fig. 14 we show examples of time histories of the slow dynamic z(t) obtained by numerical simulation of system (11) for some values of ω picked from Fig. 13 using dde23 [36]. Figures 14a, c depict the slight modulation of the QP vibration corresponding to ω = 1.4 (Fig. 13a) and ω = 1.2 (Fig. 13c). In.

(13) Author's personal copy Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems. Fig. 10 The frequency-locked area and on the modulation amplitude vibration around the 2:1 slow resonance; τd = π/4, λ1 = 0.01 and λ2 = 0 (a) Ω = 0 and (b) Ω = 100. Fig. 11 Zoom of the synchronization area of Fig. 10. Figs. 14b, d is shown the frequency-locked response corresponding to ω = 2 (Fig. 13a) and ω = 1.48 (Fig. 13b) presented by the unique circles located at the resonance. Finally, analytical approximations of the periodic solution of the system (24) is given by (29) where ϕ is given from (30) by ϕ=−. k1 k4 (8ω2 k32 + h2 ) t 4ωk2 k3 8ω2 k32 − h2. (32). and an explicit expression of the QP response of the slow dynamic (11) is written as ω ω t + v(t) sin t (33) z(t) = u(t) cos 2 2 Figure 15 compares the analytical result (29) (solid line) and the results obtained by numerical simulation (crossed line) of the slow flow (24) in the delayed and undelayed cases and in the absence and the presence of the FHE showing good agreement..

(14) Author's personal copy M. Hamdi, M. Belhaq. Fig. 12 The frequency-locked area and the modulation amplitude vibration near primary resonance; parameter values as in Fig. 10 except that λ2 = 0.8 (a) Ω = 0, (b) Ω = 100. Fig. 13 Zoom of the synchronization area of Fig. 12. In Fig. 16 we compare between the analytical result of the QP solutions (33) (solid line) and the results obtained by numerical integration (doted line) of the slow dynamic (12) in the presence of time delay and the FHE.. 5 Conclusions We have studied the effect of time-delayed feedback on the QP oscillation envelopes and frequency-locking area near primary resonance in a MvdPD oscillator. subjected to a FHE. An analytical approach, called TSPA, based on the direct separation of motions and two stages of the MSM has been performed to obtain approximate analytical expressions for stationary periodic vibrations, QP oscillation envelopes as well as frequency-locking area. The results show that, in the presence of FHE, an increase of the nonlinear feedback gain decreases drastically the amplitude of the resonance and introduces a softening effect (Figs. 7a, b), while an increase of the time delay hardens the system (Figs. 7b, c, d)..

(15) Author's personal copy Quasi-periodic oscillation envelopes and frequency locking in rapidly vibrated nonlinear systems. Fig. 14 Examples of time histories of the slow dynamic z(t) by numerical integration for parameter values as in Fig. 12 except that (a) Ω = 0, ω = 1.4, (b) Ω = 0, ω = 2, (c) Ω = 100, ω = 1.2 and (d) Ω = 100, ω = 1.48. Regarding the QP oscillations, the results shown that increasing the nonlinear feedback gain decreases significantly the amplitude and the envelopes of the QP vibrations in the whole frequency range (Fig. 12). The decreasing of the QP oscillation envelopes provides a quasi synchronization in the system over the whole frequency range around the resonance. This result can be of importance in mechanical applications involving parametric and self-excited vibrations in which a frequency-locked response in a large frequency range is required. It was also concluded that the quasi synchronization area is reduced to a very small region at the resonance; see the sin-. gle circles at the resonance in the zoomed figures (Figs. 11, 13). An important contribution of this paper is the implementation of the TSPA method to produce analytical approximations of QP oscillation envelopes in harmonically forced and self-excited weakly nonlinear oscillator with time delay. Although the method was applied to a specific oscillator, the obtained analytical results approximating the quasi-periodic oscillations envelopes are new. It should be noticed that the TSPA method can be extended to other forced nonlinear oscillators and my be applied to specific mechanical/physical systems evolving in a dynamic in-.

(16) Author's personal copy M. Hamdi, M. Belhaq. Fig. 15 Analytical approximation (solid line) of periodic solution, (29), and numerical integration (crossed line) of the slow flow (24) near the primary resonance. Fig. 16 Analytical approximation (solid line) of QP solution, (33), and numerical integration (doted line) of the slow dynamic (12) for λ1 = 0.01, λ2 = 0.05, τd = π/4, Ω = 100 and ω = 1.22. cluding external/parametric forcing and self-excitation coupled with weak nonlinearities and active control devices.. References 1. Tondl, A.: On the interaction between self-excited and parametric vibrations. In: Monographs and Memoranda, vol. 25. National Research Institute for Machine Design, Prague (1978). 2. Schmidt, G.: Interaction of Self-Excited Forced and Parametrically Excited Vibrations. In: The 9th International Conference on Nonlinear Oscillations, vol. 3. Application of the Theory of Nonlinear Oscillations. Naukowa, Dumka (1984) 3. Belhaq, M.: Numerical study for parametric excitation of differential equation near a 4-resonance. Mech. Res. Commun. 17(4), 199–206 (1990) 4. Szabelski, K., Warminski, J.: Self excited system vibrations with parametric and external excitations. J. Sound Vib. 187(4), 595–607 (1995) 5. Szabelski, K., Warminski, J.: The nonlinear vibrations of parametrically self-excited system with two degrees of freedom under external excitation. Nonlinear Dyn. 14, 23–36 (1997) 6. Warminski, J.: Synchronisation effects and chaos in van der Pol–Mathieu oscillator. J. Theor. Appl. Mech. 39(4), 861– 884 (2001) 7. Warminski, J., Balthazar, J.M.: Vibrations of a parametrically and self-excited system with ideal and non-ideal energy sources. J. Braz. Soc. Mech. Sci. Eng. 25, 413–420 (2003) 8. Belhaq, M., Fahsi, A.: Higher-order approximation of subharmonics close to strong resonances in the forced oscillators. Comput. Math. Appl. 33(8), 133–144 (1997) 9. Yano, S.: Analytic research on dynamic phenomena of parametrically and self-exited mechanical systems. Ing.Arch. 57, 51–60 (1987) 10. Yano, S.: Considerations on self- and parametrically excited vibrational systems. Ing.-Arch. 59, 285–295 (1989) 11. Abouhazim, N., Belhaq, M., Lakrad, F.: Three-period quasi-periodic solutions in self-excited quasi-periodic Mathieu oscillator. Nonlinear Dyn. 39, 395–409 (2005) 12. Pandey, M., Rand, R.H., Zehnder, A.: Frequency locking in a forced Mathieu–van der Pol–Duffing system. Nonlinear Dyn. 54, 3–12 (2008) 13. Pandey, M., Rand, R.H., Zehnder, A.: Perturbation analysis of entrainment in a micromechanical limit cycle oscil-.

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