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Nonlinear transition dynamics in a time-delayed vibration isolator under combined harmonic and stochastic excitations

This content has been downloaded from IOPscience. Please scroll down to see the full text. J. Stat. Mech. (2017) 043202 (http://iopscience.iop.org/1742-5468/2017/4/043202) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 202.118.249.121 This content was downloaded on 06/04/2017 at 08:54 Please note that terms and conditions apply.

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J

ournal of Statistical Mechanics: Theory and Experiment

PAPER: Classical statistical mechanics, equilibrium and non-equilibrium

Tao Yang and Qingjie Cao Centre for Nonlinear Dynamics Research, School of Astronautics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China E-mail: [email protected] and [email protected] Received 9 October 2016, revised 25 November 2016 Accepted for publication 25 November 2016 Published 5 April 2017 Online at stacks.iop.org/JSTAT/2017/043202 https://doi.org/10.1088/1742-5468/aa50dc

Abstract. Based on the quasi-zero stiness vibration isolation (QZS-VI)

system, nonlinear transition dynamics have been investigated coupled with both time-delayed displacement and velocity feedbacks. Using a delayed nonlinear Langevin approach, we discuss a new mechanism for the transition of a vibration isolator in which the energy originates from harmonic and noise excitations. For this stochastic process, the eective displacement potential, stationary probability density function and the escape ratio are obtained. We investigate a variety of noise-induced behaviors aecting the transitions between system equilibria states. The results indicate that the phenomena of transition, resonant activation and delay-enhanced stability may emerge in the QZS-VI system. Moreover, we also show that the time delay, delay feedback intensities, and harmonic excitation play significant roles in the resonant activation and delay-enhanced stability phenomena. Finally, a quantitative measure for amplitude response has been carried out to evaluate the isolation performance of the controlled QZS-VI system. The results show that with properly designed feedback parameters, time delay and displacement feedback intensity can play the role of a damping force. This research provides instructive ideas on the application of the time-delayed control in practical engineering.

Keywords: dynamical processes

© 2017 IOP Publishing Ltd and SISSA Medialab srl

1742-5468/17/043202+25$33.00

J. Stat. Mech. (2017) 043202



Contents 1. Introduction 2 2. Stochastic delayed model of vibration isolation 3

4. Amplitude response and comparative analysis 16 4.1. Amplitude equations and stationary solutions. . . . . . . . . . . . . . . . . . . 16 4.2. Comparative analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5. Concluding remarks 22 Acknowledgments 23 References

23

1. Introduction With the increased interest in nonlinear passive vibration isolation systems, many research studies are currently focused on developing technologies to enhance their ecacy, especially for dierent designs of structures [1–3] and control methods [4–6]. One of the control methods for the isolation of vibration is time-delayed active control. In the past two decades, much attention has been paid to studying the time delay problem in the control community [7–12]. Time delay is inevitable in nature, technology, and society due to finite signal transmission times, switching speeds, and memory eects. Therefore, studying the eects of time delay not only plays a significant role in understanding the physical mechanism of control systems in the real world, but is also of significant interest in the field of nonlinear science. Time delay eects on controlled systems subject to deterministic excitation have been studied extensively [13–17]. Of special importance is the development of isolators that can scavenge vibration eciently from non-stationary and stochastic excitation sources. This has recently become a more pressing issue, especially with the knowledge that such excitations represent a large segment of the environmental vibratory energy sources. For controlled systems with time delays excited by Gaussian white noise, Di Paola et al [18, 19] have studied the eects of time delay on the controlled linear and nonlinear systems by using an approach based on the Taylor expansion of the control force and another approach to finding an exact stationary solution. Using the stochastic averaging method, the response and stability of quasi-integrable Hamiltonian systems with delayed feedback control under Gaussian white noise excitation have been studied [20–22]. Accordingly, https://doi.org/10.1088/1742-5468/aa50dc

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3. Transition dynamics in a vibration isolator 6 3.1. The eective potential of the FPK equation. . . . . . . . . . . . . . . . . . . . 6 3.2. Noise- and delay-induced transitions.. . . . . . . . . . . . . . . . . . . . . . . . 8 3.3. The escape rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10


2. Stochastic delayed model of vibration isolation The mechanical model adopted in this study represents the dynamics of a QZS-VI system, which, as shown in figure 1, is the extension of the classical smooth and discontinuous oscillator [25]. The system consists of a lumped mass M, a vertical linear spring https://doi.org/10.1088/1742-5468/aa50dc

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a natural or artificial time-delayed system is always influenced by weak or strong noise. Hence, the study of noise-induced dynamics in time-delayed systems has received increasing attention recently. As mentioned above, the excitation of systems is a purely harmonic excitation or purely random noise. However, many mechanical and structural systems are often subjected to both stochastic and harmonic excitations [23, 24]. So far, to the authors’ knowledge, no work on time-delayed vibration isolators under combined harmonic and stochastic excitations is available. During the last couple of years, many researchers have also purposefully exploited the introduction of stiness-type nonlinearities into the vibration isolator’s design. The design concepts of quasi-zero stiness and high-static-and-low-dynamic stiness have been proposed and studied as a hot topic of microamplitude vibration control in recent years, which is essentially the combination of positive and negative stiness elements. The positive stiness components provide a high load capacity when the isolated objective rests, while the assembly of the positive and negative stiness components works together to provide a low dynamic stiness when the isolated objective moves under the ambient disturbances. The work in this paper is based on the research of the onedirection quasi-zero stiness vibration isolation (QZS-VI) by using the negative stiness given by a smooth and discontinuous oscillator [25] and a supporting positive stiness which indicate the high-static-and-low-dynamic stiness [26–28]. Because of the highstatic-low-frequency property, the one-direction QZS-VI systems have been extensively studied in the literature [25–35, 36] to provide a better working environment for human beings or to protect precision instruments. Inspired by this, the current paper deals with the nonlinear transition dynamics in a QZS-VI dynamical system, coupled with both time-delayed displacement and velocity feedbacks under combined harmonic and real noise excitations in which we try to unveil how the noise and time delay induce transition behaviors, and the roles played by the noise and time delay which cause complex dynamics in the time-delayed QZS-VI dynamical system. In particular, we identify the possible emergence of transitions, resonant activation and delay enhanced stability, and the eects of delay feedback intensities and noise on them. This paper is organized as follows. In section 2, a QZS isolator with time-delayed active control under combined harmonic and real noise excitations is introduced and modeled. Section 3 formulates the problem in the Itô stochastic sense and presents the Fokker–Planck–Kolmogorov (FPK) equation governing the evolution of the transition stationary probability density function (SPDF). We analyze the noise- and delay-induced nonlinear transition dynamics in the QZS-VI system in terms of the eective displacement potential, the SPDF of displacement and the escape ratio. In section 4, the stochastic averaging method is applied to obtain the SPDF for amplitude in a time-delayed QZS isolator, and the eects of noise, time delay feedback and the damping coecient on amplitude SPDF are studied. Finally, section 5 presents the important conclusions.


(with a stiness k1, to support the static load), and a pair of oblique linear springs (with a stiness, k2 ). c is the damping coecient. A time-delayed displacement and velocity feedback control force Fc = K1x (t − τ ) + K2x˙ (t − τ ) is introduced to the system. Applying harmonic excitation and considering environmental base excitation in the primary system, the non-dimensional vibration equation governing the motion can be expressed as

⎛ ⎞ 1 ⎟⎟ = K1x (t − τ ) + K2x˙ (t − τ ) + Fth(t ) + F cos ωt . (1) x¨ + 2cx˙ + x + γx ⎜⎜1 − ⎝ x 2 + α2 ⎠ In this equation, x is the relative displacement of the mass. The stiness ratio γ, geometrical arrangement ratio α and damping coecient c are constants. As the value of 0 < α < 1, the smooth and discontinuous oscillator will possess a negative stiness, which can counteract the positive stiness to generate the QZS when it is coupled with the vertical spring [26–28]. K1 and K2 are feedback intensities and τ > 0 is the feedback time delay. Fth(t ) is the environmental base excitation and F cos ωt is an externally harmonic forcing. The response of the system (1) is a stochastic process, which is non-Markovian due to the time delay contained in the feedbacks. A multiple scale expansion method [37, 38] is employed to derive the stationary probability distribution of the Van der Pol-type oscillator subjected to Gaussian noise sources. We hypothesize that there are two main time scales: one is a one-order time scale t and the other is a time scale T = ε 2t (ε > 0), which is slower than the former. Stochastic contributions are supposed to average away at t or even faster; thus, the randomness remains only at the slower time scale T. Thus, the solution of equation (1) can be written as x (t , T ) = εA(T ) cos(ωt ) − εB (T ) sin(ωt ), so that x˙ (t , T ) = −εωA(T ) sin(ωt ) − εωB (T ) cos(ωt ), where ω is the dominant frequency of the oscillation. A(T) and B(T) are two stochastic processes for which only the evo lution on the slow scale T is retained. When the time delay is considered, we get

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Figure 1. Physical model of a QZS-VI dynamical system coupled with both timedelayed displacement and velocity feedbacks. 1-sensor, 2-controller, 3-actuator.


⎧x (t − τ, T − ε 2τ ) = εA(T − ε 2τ ) cos[ω(t − τ )] − εB (T − ε 2τ ) sin[ω(t − τ )], ⎨ (2) ⎩x˙ (t − τ, T − ε 2τ ) = −εωA(T − ε 2τ ) sin[ω(t − τ )] − εωB (T − ε 2τ ) cos[ω(t − τ )]. Since the time delay τ is finite, A(T − ε 2τ ) ≈ A(T ), B (T − ε 2τ ) ≈ B (T ). Then, equations (2) reduce to ⎧ ⎪x (t − τ ) ≈ x cos(ωτ ) − x˙ sin(ωτ ) , ⎨ ω ⎪x˙ (t − τ ) ≈ xω sin(ωτ ) + x˙ cos(ωτ ). ⎩

(3)

⎛ x¨ + ξx˙ + ω 2x + γx ⎜⎜1 − ⎝

with

⎞ ⎟⎟ = Fth(t ) + F cos ωt 2 2 ⎠ x +α 1

⎧ ⎪ ξ = 2c + K1 sin(ωτ ) − K2 cos(ωτ ), ⎨ ω ⎪ ω 2 = 1 − K cos(ωτ ) − K ω sin(ωτ ). ⎩ 1 2

(4)

(5)

Note that delay feedback still aects the system (4), with equations (5) considered. Accordingly, equation (4) is not equivalent to the system without a time delay. Without excitations and delay feedback control, let c = 0 and F = 0, the equilibria of the system (1) with QZS properties for 0 < α < 1 can be written as ⎛ ⎛ ⎞ ⎛ ⎞ 2 ⎛ γ ⎞2 γ ⎞ ⎜ ⎟ ⎜ 2 2 xA ⎜ − α , 0 , x 0(0, 0), xB − ⎜ ⎟ − α , 0⎟⎟. ⎜ ⎝ 1 + γ ⎟⎠ ⎟ ⎜ γ + 1 ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

(6)

The Jacobian of the QZS-VI system is

⎡ 0 1⎤ ⎢ ⎥ 2 γα J = ⎢−(1 + γ ) + ⎥. 0 3 2 2 ⎢⎣ (x + α )2 ⎥⎦

(7)

Thus, the Jacobian according to equilibria xA, xB and x0 are obtained, respectively, ⎡ 0 ⎢ JxA, xB = ⎢−(1 + γ ) + ⎢ ⎢⎣

γα

2

( )

3 γ 1+γ

1⎤ ⎡ 0 1⎤ ⎥ 0 ⎥ , Jx 0 = ⎢−(1 + γ ) + γ 0 ⎥ , ⎢⎣ ⎥ ⎥ α ⎦ ⎥⎦

(8)

with the characteristic equations of the system written as β 2 + (1 + γ ) −

γα2

( )

3 γ 1+γ

= 0,

γ λ2 + (1 + γ ) − = 0. α


(9)

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By substituting equations (3) into equation (1), one finds


The eigenvalues of the linearization matrix with σ = 0 and F = 0 at the equilibria xA and xB can be obtained as (1 + γ )3α2 β . 1,2(A, B ) = ± −(1 + γ ) + γ2

(10)

In the same way, we can obtain the eigenvalues of the linearization matrix with σ = 0 and F = 0 at the equilibrium x0 as λ 1,2(0) = ± −(1 + γ ) +

γ . α

(11)

In this section, based on the analysis results of the eective displacement potential, the SPDF of displacement and the escape ratio, we discuss the eects of the parameters of noise, time delay feedback, and harmonic forcing on the stationary and transient properties of the QZS-VI system to indicate the possible emergence of three well-known noise- and delay-induced phenomena, i.e. transitions, resonant activation and delay enhanced stability. 3.1. The eective potential of the FPK equation

Throughout this study, the environmental base excitation, Fth(t ), is assumed to be a physical Gaussian process with a very small correlation time which approaches zero. In such a case, Fth(t ) can be approximated by a Gaussian white noise process such that Fth(t ) = 0, Fth(t )Fth(t ′) = 2σδ (t − t ′),

(12)

characterizes averaging with respect to the noise Fth(t ), σ is the intensity of the randomness, and δ is the Dirac-delta function. To generate the response statistics associated with the stochastic dynamics (4), we further express it in the Itô stochastic form as [39] dX (t ) = f (X , t ) + g (X , t )Γ(t ), dt

(13)

where X = (x 1, x 2)T ≡ (x , x˙ )T , Γ(t ) is a Gaussian white noise. The QZS-VI dynamical system (4) can be written in a two-dimensional system as follows: ⎧x˙ 1 = x 2, ⎪ ⎪ ⎛ ⎨ x˙ = −ξx − ω 2x − γx ⎜1 − 2 1 1⎜ ⎪ 2 ⎪ ⎝ ⎩

⎞ ⎟ + Fth(t ) + F cos ωt . 2 2 ⎟ x1 + α ⎠ 1

(14)

Furthermore ω 2 ≈ 1 corresponds to the frequency in the harmonic limit. This yields


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3. Transition dynamics in a vibration isolator


⎧x 2 ⎫ ⎪ ⎪ ⎛ ⎞ 0 1 f (X , t ) = ⎨ (15) ⎜1 − ⎟ + F cos ωt ⎬, g (X , t ) = 1 . x x x − ξ − − γ 2 1 1 ⎪ ⎪ ⎜ 2 2 ⎟ ⎪ ⎪ x1 + α ⎠ ⎝ ⎩ ⎭ The solution of (13) is determined by the evolution of the transition probability density function, Q(X,t) , which, in turn, is governed by the following FPK equation corre sponding to equation (13) [39]:

{}

2

2

∂2 ∑ ∑ ∂x x [(MggT )ij Q (X , t )], i j i=1 j =1

⎡0 0 ⎤ M = ⎢⎣ 0 σ ⎥⎦ .

(16)

(17)

With the knowledge of f(X,t) and g(X,t), the FPK equation reduces to ∂ ∂ Q (X , t ) = − x 2Q (X , t ) ∂t ∂x 1 ⎛ ⎛ ∂ ⎜ − −ξx 2 − x 1 − γx 1⎜⎜1 − ∂x 2 ⎜ ⎝ ⎝ +

⎞ ⎞ ⎟ + F cos ωt ⎟Q (X , t ) ⎟ ⎟ x 12 + α2 ⎠ ⎠ 1

∂2 Q ( X , t ). ∂x 22

(18)

From equation (18), a joint transition SPDF Qst(x 1, x 2, t ) for displacement x1 and velocity x2 can be derived as Q st(x 1, x 2, t ) = N exp[−UFP(x 1, x 2, t )],

(19)

where N is a normalization constant and UFP(x 1, x 2, t ) is the eective potential energy, which can be expressed exactly as: U FP(x 1, x 2, t ) =

⎞ ξ ⎛1 2 1 + γ 2 ⎜ x + x 1 − γ x 12 + α2 − x 1F cos ωt ⎟. 2 ⎠ 2 σ⎝2

(20)

The eective potential energy is included in our QZS-VI system to simulate the presence of a barrier to be surmounted during the translocation dynamics. It should be pointed out here that, because the value of the periodic function in each cycle is the same, we only need to analyze a cycle of harmonic forcing. Along with the time evo lution in a cycle, ωt changes from 0 to 2π and −1 ⩽ cos ωt ⩽ 1, and therefore the values of the harmonic forcing F cos ωt distributes in [−F,F], i.e. in the first half cycle, F cos ωt changes from F to 0 and in the second half cycle, F cos ωt changes from 0 to −F, so that we can imply that the use of excitation magnitude F takes a positive value to represent harmonic forcing in the first half cycle ignoring the harmonic forcing change with time, and the use of excitation magnitude F takes a negative value to represent periodic force in the second half cycle. https://doi.org/10.1088/1742-5468/aa50dc

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2 ∂ ∂ 1 Q (X , t ) = − ∑ [ fi (X , t )Q (X , t )] + ∂t 2 i = 1 ∂x i Q (∞, t ) = Q (−∞, t ) = 0, where


In what follows, the eects of each parameter on the stationary and transient properties are investigated. Noise and time delay present in the system can induce transitions between two alternative equilibria states; a quantity of interest is the mean escape time it takes the system to change its regime from the equilibrium state xA to xB. 3.2. Noise- and delay-induced transitions

The eect of stiness ratio γ, geometrical arrangement ratio α and damping coecient c on the eective displacement potential UFP(x 1) and the SPDF Qst(x 1) of the displacement x1 is depicted in figure 2, respectively. In the left panels in figure 2, we present the eective displacement potential UFP(x 1) as a function of the displacement x1. We show that by varying γ from low to high values (see figure 2(a)), the system can pass from a monostable equilibrium state x0 to a region of bistability (i.e. the coexistence of equilibrium states xA and xB). For γ = 0.5, the region of bistability is distinguished by the presence of two local minima in the eective displacement potential UFP(x 1); the left and right minima correspond to equilibrium states xB and xA, respectively. It is found from figure 2(a) that the depth of the potential minima at x = xA and x = xB increases and the potential is enhanced as the γ increases from 0.5–0.9. For increasing α, the depth of the potential minima at x = xA and x = xB decrease until they disappear (see figure 2(b)). From figure 2(c), it is seen that the depth of the potential minima at x = xA and x = xB decreases and the potential is reduced as c increases. In short, a physical explanation of the mechanisms for noise-driven motion is derived from the eective potential of the FPK equation. On the other hand, in stochastic models, peaks of the SPDF Qst(x 1) correspond to attractors, and troughs correspond to repellers. Moreover, https://doi.org/10.1088/1742-5468/aa50dc

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Figure 2. Eect of stiness ratio γ, geometrical arrangement ratio α and damping coecient c on (a)–(c) the eective displacement potential UFP(x 1) and (d)–(f ) the SPDF Qst(x 1) of the displacement x1. (a), (d) α = 0.1, c = 0.01; (b), (e) γ = 0.9, c = 0.01; (c) ,(f ) γ = 0.9, α = 0.1. Other parameter values are τ = 0.9, K1 = 0.9, K2 = 0.1, σ = 0.1 and F = 0.1.


an equilibrium point is more stable (high resilience) if the SPDF peak is large in comparison with another equilibrium point which is less stable (low resilience) as the SPDF Qst(x 1) peak is small. The eect of the three values of the γ, α and c on the SPDF Qst(x 1) is shown in figures 2(d)–(f ). We can see that the structure of the SPDF Qst(x 1) changes from unimodal to bimodal when γ is increased (see figure 2(d)). Conversely, for increasing α the structure of the SPDF Qst(x 1) changes from bimodal to unimodal (see figure 2(e)). As the value of c increases, two peaks develop, and the height of the peaks increases, respectively. Namely, the structure of the SPDF Qst(x 1) switches from the unstable equilibrium state x0 to stable equilibrium states xA and xB as the value of c increases (see figure 2(f )). In other words, the eect of stiness ratio γ, geometrical arrangement ratio α and damping coecient c on the eective potential UFP(x 1) and the SPDF Qst(x 1) is consistent. Figure 3 shows the variation of the eective displacement potential UFP(x 1) and the SPDF Qst(x 1) versus the displacement x1 for dierent values of time delay τ, time-delayed displacement feedback intensity K1 and velocity feedback intensity K2, respectively. On one hand, when we increase τ or K1 by other fixed parameters (see figures 3(a) and (b)), the depth of the potential minima at x = xA and x = xB increases and the potential is enhanced as the value of τ or K1 increases. But for increasing K2, the depth of the potential minima at x = xA and x = xB decreases and the potential is reduced as the value of K2 increases (see figure 3(c)). On the other hand, the eect of the three values of τ, K1 and K2 on the SPDF Qst(x 1) is shown in figures 3(d)–(f ). In the bistable region, as the value of τ or K1 increases (figures 3(d) and (e)), the height of the two peaks (corresponding to equilibrium states xA and xB) increases, respectively. From figure 3(f ), it is seen that the height of the two peaks decreases. These findings are also consistent with the results of figures 3(a)–(c). https://doi.org/10.1088/1742-5468/aa50dc

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Figure 3. Eect of time delay τ, delay feedback intensities K1 and K2 on (a)– (c) the eective displacement potential UFP(x 1) and (d)–(f ) the SPDF Qst(x 1) of the displacement x1. (a), (d) K1 = 0.9, K2 = 0.1; (b), (e) τ = 0.9, K2 = 0.1; (c), (f ) τ = 0.9, K1 = 0.9. Other parameter values are γ = 0.9, α = 0.1, c = 0.01, σ = 0.1 and F = 0.1.


To study the eect of noise intensity σ and excitation magnitude F on the stationary properties, the response cures for varying parameters σ and F are shown in figure 4. The eective displacement potential UFP(x 1) as a function of displacement x1 is shown in figure 4(a) for dierent values of σ and figures 4(b) and (c) for dierent values of F. As the values of σ increase the depth of the potential minima at x = xA and x = xB decreases and the potential is reduced. From figure 4(b), the depth of the potential minima at x = xA increases and x = xB decreases with γ = 0.9. But for γ = 0 (i.e. the linear isolator without the horizontally auxiliary springs) there is only one maximum and the position of the maximum changes from left to right as the values of F change from −0.2 to 0.2. Furthermore, figure 5 shows the mean displacement x 1 st of the QZS-VI system versus stiness ratio γ, geometrical arrangement ratio α, damping coecient c, time delay τ, time-delayed displacement feedback intensity K1 and velocity feedback intensity K2, noise intensity σ and excitation magnitude F, respectively. Here x 1 st is defined as +∞

x 1 st = ∫ x 1Qst(x 1)dx 1. It is shown that the mean displacement x 1 st increases with −∞ increasing γ, c, τ, K1 and F, but decreases with increasing α, K2 and σ. Finally, it is necessary to get an insight into the joint transition SPDF Qst(x 1, x 2). In figure 6 we show a three-dimensional view of the joint transition SPDF Qst(x 1, x 2) as functions of displacement x1 and velocity x2. 3.3. The escape rate

In this section, we are more interested in how the escape from the equilibrium state xA to xB is aected by the environmental base excitation and time delay feedback in the QZS-VI system. The synergistic action of noise, time delay and system can also be https://doi.org/10.1088/1742-5468/aa50dc

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Figure 4. Eect of noise intensity σ and excitation magnitude F on (a)–(c) the eective displacement potential UFP(x 1) and (d)–(f ) the SPDF Qst(x 1) of the displacement x1. (a), (d) γ = 0.9, F = 0.1; (b), (e) γ = 0.9, σ = 0.1; (c), (f ) γ = 0, σ = 0.1. Other parameter values are α = 0.1, c = 0.01, τ = 0.9, K1 = 0.9 and K2 = 0.1.


studied from the perspective of Kramers rate. In the stochastic process of equation (1), the noise-driven motion transits at a certain rate, whose value is given by the wellknown Kramers rate [40]. The exact expression of the escape rate is given by [41]. We extended this approach to investigate the escape rate of the QZS-VI system from the equilibrium state xA escape to xB. In essence, one begins with Kolomogorov’s backward equation which is equivalent to (20). This ultimately leads to the escape rate RAB for R AB = R xA→ xB =

β1β2 2π

−

λ1 exp [UFP(xA, t ) − UFP(x 0, t )] , λ2

(21)

In the same way, the probability escape rate RBA can be written as R BA = R xB → xA =

β1β2 2π

−

λ1 exp [UFP(xB, t ) − UFP(x 0, t )] , λ2

(22)

where β1,2 and λ1,2 are given by equations (10) and (11), respectively. Based on equations (21) and (22), the eect of noise, delay feedback and harmonic forcing on the escape rates can be analyzed, respectively. Figure 7 represents the escape rate RAB as a function of noise intensity σ for dierent values of delay feedback intensities K1, K2 and excitation magnitude F. We can see from figures 7(a) and (c) that RAB decreases slowly as K1 or F increases, and RAB https://doi.org/10.1088/1742-5468/aa50dc

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Figure 5. The mean displacement x 1 st as functions of (a) stiness ratio γ, (b) geometrical arrangement ratio α, (c) damping coecient c, (d) time delay τ, (e) delay feedback intensities K1, (f) K2, (g) noise intensity σ and (h) excitation magnitude F, respectively. The parameter values are γ = 0.9, α = 0.1, c = 0.01, τ = 0.9, K1 = 0.9, K2 = 0.1, σ = 0.1, F = 0.1.


decreases quickly as σ increases when the values of K1 −0.5. It illustrates that the escape from the equilibrium state xA to xB is easy as the delay feedback K1 −0.5, and the stability of the equilibrium state xA is enhanced by noise. In figure 7(b), RAB increases slowly as K2 increases, and RAB increases quickly as σ increases when the values of K2 0.3. This illustrates that the escape from the equilibrium state xA to xB is dicult as the delay feedback K2 0.3, and the stability of the equilibrium state xA is weakened by noise. In figure 8, we plot the escape rate RAB as a function of time delay τ for dierent values of excitation magnitude F, delay feedback intensities K1, K2 and noise intensity σ. Figure 8(a) shows that with an increase in time delay τ, the escape rate RAB first increases, followed by a decrease after passing through a maximum for an optimal value of τ. Thus, a turnover behavior of RAB with the variation of time delay is observed. The turnover is a hallmark for the resonant activation (RA) phenomenon. Physically, the above-mentioned meaningful modification of the RA phenomenon can be attributed to the energy barrier fluctuation and the memory eect caused by the time delay. In addition, it is worth noting that, in a strict sense, the RA phenomenon occurring in our system should be named ‘delay resonant activation’ to distinguish the stochastic resonant activation [42–45]. Figure 8(a) also suggests that the RA maximum of RAB versus https://doi.org/10.1088/1742-5468/aa50dc

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Figure 6. 3D views of the resulting joint transition SPDF Qst(x 1, x 2) of displacement x1 and velocity x2 for the parameters γ = 0.9, α = 0.1, c = 0.01, τ = 0.9, K1 = 0.9, K2 = 0.2, σ = 0.2, F = 0.1.


τ decreases as F increases, i.e. excitation magnitude F induced barrier fluctuation is not conducive to the RA phenomenon. This result implies that the increase in τ leads to an ascension of the escape rate and reduces the likelihood of the escape to the xB state, but an increase in F leads to a drop of the escape rate and enhances the likelihood of the escape to the xB state. The eect of the delay feedback intensities K1, K2 and noise intensity σ on the RA phenomenon is plotted in figures 8(b)–(d) with F = −0.9. One can see from figures 8(b) and (c) that the peak height of the escape rate RAB as a function of time delay τ increases as the values of K1 or K2 increase, but decreases as the values of σ increase (see figure 8(d)). Namely, delay feedback intensities K1 and K2 can accelerate the escape to the xB state, and the noise intensity σ can slow down the escape. This implies that the RA eect can be strengthened by enhancing K1 or K2, and can also be weakened by enhancing σ. More interestingly, for the weak negative and positive excitation magnitude case, the escape rate RAB as a function of time delay τ shows a non-monotonic behavior with the presence of a minimum (see figure 9(a)). This is a clear signature of the delay enhanced stability (DES) phenomenon [46], which is dierent to the noise-enhanced stability case [47–53]. The DES eect implies that the time delay can stabilize a fluctuation in such a way that the system remains in this state for a longer time than in the absence of time delay. In the present situation, the occurrence of the DES eect signifies that the stability of the equilibrium state xA can be enhanced by the time delay. Thus, the system https://doi.org/10.1088/1742-5468/aa50dc

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Figure 7. The escape rate RAB as a function of noise intensity σ for dierent delay feedback intensities (a) K1, (b) K2 and (c) excitation magnitude F. (a) K2 = 0.1, F = 0.1; (b) K1 = 0.5, F = 0.1; (c) K1 = 0.5, K2 = 0.1. Other parameter values are γ = 0.9, α = 0.1, c = 0.01, τ = 0.5.


can stay in the equilibrium state xA for a longer time due to the role of the time delay feedback control through the DES mechanism. Figure 9(a) also suggests that the DES minimum of RAB versus τ decreases as F increases, i.e. excitation magnitude F induced barrier fluctuation is conducive to the DES phenomenon. This result implies that the increase in τ or F leads to a drop of the RAB and enhances the likelihood of the escape to the xB state. Figures 9(b)–(d) show the eect of delay feedback intensities K1, K2 and noise intensity σ on the DES phenomenon with F = 0.1. It seems that the minimum of RAB versus τ decreases as the values of K1 or K2 increase (see figures 9(b) and (c)), but increases as the values of σ increase (see figure 9(d)). In other words, delay feedback intensities K1 and K2 can slow down the escape to the xB state, and noise intensity σ can accelerate the escape. This implies that the DES eect can be strengthened by enhancing K1 or K2, and can also be weakened by enhancing σ. To outline the probability escape rate RBA, we show in figures 10–12 the escape rate RBA as a function of noise intensity σ or time delay τ for dierent values of harmonic forcing and delay feedback intensity. Figures 10–12 describe the same type of analysis of RAB. Also, in this case, the RA and DES eects can be strengthened by enhancing the displacement or velocity feedback intensity, and can be weakened by enhancing noise intensity, though the opposite is the case for the eect of the excitation magnitude on the escape rates RBA and RAB. https://doi.org/10.1088/1742-5468/aa50dc

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Figure 8. The escape rate RAB as a function of time delay τ for dierent (a) excitation magnitude F, delay feedback intensities (b) K1, (c) K2 and (d) noise intensity σ. (a) K1 = 0.5, K2 = 0.1, σ = 0.5; (b) K2 = 0.1, σ = 0.1, F = −0.9; (c) K1 = 0.5, σ = 0.1, F = −0.9; (d) K1 = 0.5, K2 = 0.1, F = −0.9. Other parameter values are γ = 0.9, α = 0.1, c = 0.01.


Synthesizing the above analysis, to obtain insight into the physics of the nonlinear transition dynamics, we focus our investigation on the probability density function of the escape time. According to Kramers’ law [54, 55], the probability density function of the escape time is thus given by P i (s ) = Ri exp[−Ris ], for i = AB, and BA,

(23)

where the escape rate Ri is given by equations (21) and (22), respectively. This result is for the escape time distribution (ETD) for each well. When the param eters γ = 0.9, α = 0.1, c = 0.01, F = 0.1 are fixed, the analytical results of the ETD PAB(s) are shown in figure 13 for dierent noise intensities σ, time delay τ, delay feedback intensities K1 and K2, respectively. Figure 13 shows that the PAB(s) decays exponentially as time s increases. Moreover, as the values of the noise intensity σ (see figure 13(a)) or delay feedback intensity K2 (see figure 13(d)) increased, the PAB(s) increased monotonously. For the increasing delay feedback intensity K1 (see figure 13(c)), a monotonic decay of PAB(s) is observed. However, figure 13(b) shows that the PAB(s) passes through a minimum with varying time delay τ, which demonstrates that when the time delay τ satisfies a sort of matching condition, the DES phenomenon might occur. https://doi.org/10.1088/1742-5468/aa50dc

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Figure 9. The escape rate RAB as a function of time delay τ for dierent (a) excitation magnitude F, delay feedback intensities (b) K1, (c) K2 and (d) noise intensity σ. (a) K1 = 0.5, K2 = 0.1, σ = 0.5; (b) K2 = 0.1, σ = 0.1, F = 0.1; (c) K1 = 0.5, σ = 0.1, F = 0.1; (d) K1 = 0.5, K2 = 0.1, F = 0.1. Other parameter values are γ = 0.9, α = 0.1, c = 0.01.


4. Amplitude response and comparative analysis This section is divided into two parts. In the first part, the stationary amplitude response analysis for the time-delayed QZS-VI dynamical system is shown; the second part presents a comparative analysis of the displacement and amplitude SPDFs. 4.1. Amplitude equations and stationary solutions

This section deals with an analytical investigation of the amplitude response associated with the equation of the QZS-VI dynamical system (1). In the quasiharmonic regime, assuming that the noise intensity is small, one adopts the change of variables: x (t ) = A(t ) cos[ωt + θ(t )], x˙ (t ) = −A(t )ω sin[ωt + θ(t )], φ(t ) = ωt + θ(t ). Substituting equation (24) into equation (4), it can yield the equations as ⎧ ˙ = − 1 M (A, θ ) sin φ, A ⎪ ⎪ ω ⎨ 1 ⎪ M (A, θ ) cos φ, ⎪ θ˙ = − ⎩ Aω


(24)

(25)

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Figure 10. The escape rate RBA as a function of noise intensity σ for dierent delay feedback intensities (a) K1, (b) K2 and (c) excitation magnitude F. (a) K2 = 0.1, F = 0.1; (b) K1 = 0.5, F = 0.1; (c) K1 = 0.5, K2 = 0.1. Other parameter values are γ = 0.9, α = 0.1, c = 0.01, τ = 0.5.


here M (A, θ ) = Aω 2 cos φ + ξAω sin φ − (1 + γ )A cos φ +

γA cos φ A2 cos2 φ + α2

+ Fth + F cos ωt .

(26) Averaging over the period of oscillations, we obtain the following stochastic equations for the slow (on a scale of T = 2π)-varying amplitude A(t) and phase θ

2π ⎧ 1 1 A˙ = − (2ξAω + F sin θ ) − Fth sin φ dφ, ⎪ ⎪ 2ω 2πω 0 ⎨ ⎪θ˙ = − 1 ⎛⎜−A(1 + γ ) + Aω 2 + F cos θ + Aγ N (A)⎞⎟ − 1 ⎪ ⎠ 2πAω π 2Aω ⎝ ⎩ where

∫

N (A ) = =

∫0

2π

cos2 φ A2 cos2 φ + α2

Fth cos φ dφ,

(27)

dφ

⎛ ⎛ 4 ⎜ 2 A2 2 ⎜⎜ EllipticE A α + ⎜ 2 2 A2 ⎝ ⎝ A +α

∫0

2π


⎞ ⎟⎟ − ⎠

⎛ A2 EllipticK⎜⎜ 2 2 ⎝ A +α A2 + α2 α2

⎞⎞ ⎟⎟⎟⎟. ⎠⎠

(28) 17

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Figure 11. The escape rate RBA as a function of time delay τ for dierent (a) excitation magnitude F, delay feedback intensities (b) K1, (c) K2 and (d) noise intensity σ. (a) K1 = 0.5, K2 = 0.1, σ = 0.5; (b) K2 = 0.1, σ = 0.1, F = 0.7; (c) K1 = 0.5, σ = 0.1, F = 0.7; (d) K1 = 0.5, K2 = 0.1, F = 0.7. Other parameter values are γ = 0.9, α = 0.1, c = 0.01.


By applying the stochastic averaging method [56, 57] and after integration and normalization, we obtain the following equations: ⎧ 1 ⎛ σ ⎞⎟ σ ˙ + A˙ = − ⎜2ξAω + F sin θ − W1(t ), ⎪ ⎪ 2ω ⎝ ω Aω ⎠ ⎨ ⎪ θ˙ = − 1 ⎛⎜−A(1 + γ ) + Aω 2 + F cos θ + Aγ N (A)⎞⎟ + σ W˙ (t ) 2 ⎪ ⎠ Aω 2Aω ⎝ π ⎩

(29)

where W1(t) and W2(t) represent independent normalized Wiener processes. Clearly, A˙ does not depend on θ with F = 0, and thus we can develop a probability density for amplitude A, rather than a joint density for A and θ. Let Q(A,t) denote the amplitude SPDF that the amplitude of the oscillator exactly equals A at time t. Then, from Risken [39], the delay Fokker–Planck equation of Q(A,t) corresponding to equation (4) with equation (12) can be given by ∂Q (A, t ) σ ⎞⎟ ∂ ⎛⎜ σ ∂2 ξ = − − A + Q ( A , t ) + Q (A, t ). ∂t ∂A ⎝ 2Aω 2 ⎠ ω 2 ∂A2

(30)

Within the knowledge that the amplitude SPDF Qst(A) does not change with time. Thus, the expression for Qst(A) is https://doi.org/10.1088/1742-5468/aa50dc

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Figure 12. The escape rate RBA as a function of time delay τ for dierent (a) excitation magnitude F, delay feedback intensities (b) K1, (c) K2 and (d) noise intensity σ. (a) K1 = 0.5, K2 = 0.1, σ = 0.5; (b) K2 = 0.1, σ = 0.1, F = −0.9; (c) K1 = 0.5, σ = 0.1, F = −0.9; (d) K1 = 0.5, K2 = 0.1, F = −0.9. Other parameter values are γ = 0.9, α = 0.1, c = 0.01.


Q st(A) =

⎡ NC exp ⎢ ⎣ G (A )

H (A )

⎤

∫ G (A) dA⎥⎦ ,

where NC is a normalization constant, and H(A) and G(A) are given by σ σ H (A) = −ξA + 2Aω 2 , G (A) = ω 2 .

(31)

(32)

From equations (31) and (32), the amplitude SPDF can be derived as ⎡ −ξA2ω 2 ⎤ 1 Q ( A ) = N exp + ln A⎥ . ⎢ C st ⎣ 2σ ⎦ 2

The stationary mean amplitude A A

st

=

+∞

∫−∞

st

AQst(A)dA.

(33)

can be obtained as follows: (34)

In order to see the eect of the time delay feedback and noise on the stationary properties (amplitude response) of the QZS-VI system, we have plotted the ampl itude SPDF Qst(A) as a function of amplitude A in figure 14 based on formulas (33). Furthermore, figure 15 shows the mean amplitude A st of the QZS-VI system versus time delay τ, displacement feedback intensity K1, velocity feedback intensity K2 and noise intensity σ based on formula (34), respectively. In figure 14, there is a peak; the increment of time delay τ or the displacement feedback intensity K1 will lead to the peak becoming higher (see figures 14(a) and (b)), https://doi.org/10.1088/1742-5468/aa50dc

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Figure 13. 3D plot of the ETD PAB(s) as a function of time s for dierent (a) noise intensities σ, (b) time delay τ, delay feedback intensities (c) K1 and (d) K2. (a) τ = 0.5, K1 = 0.5, K2 = 0.1; (b) σ = 0.5, K1 = 0.5, K2 = 0.1; (c) σ = 0.5, τ = 0.5, K2 = 0.1; (d) σ = 0.5, τ = 0.5, K1 = 0.5. Other parameter values are γ = 0.9, α = 0.1, c = 0.01, F = 0.1.


and the position of the peak shifts to a smaller value of amplitude A. It is shown that the mean amplitude A st decreases with increasing τ and K1 (see figures 15(a) and (b)). Remarkably, figures 15(a) and (b) also suggest that a higher time delay and displacement feedback intensity lead to the lower response of the system. In other words, the isolation performance of the time-delayed QZS-VI dynamical system can be improved by time delay and displacement feedback intensity. Figures 14(c) and (d) draw a conclusion that the peak becomes lower as the velocity feedback intensity K2 or the noise intensity σ increases, and the position of the peak shifts to a larger value of amplitude A. The mean amplitude A st increases with increasing K2 and σ (see figures 15(c) and (d)). Namely, higher velocity feedback and noise intensities lead to the stronger response of the system. To improve the isolation performance of the QZS-VI system, the results show that velocity feedback and noise intensities should be designed to be small enough. It can be found from figure 16 that the increment of the damping coecient c will also lead to the lower response of the QZS-VI system. The result is analogous to the eect of the damping coecient on the amplitude-frequency response in [27, 28] for a deterministic system without delay feedback control. The results also suggest that with properly designed feedback parameters, time delay and displacement feedback intensity can play the role of a damping force. 4.2. Comparative analysis

We have performed a detailed study on the eectiveness of time-delayed control forces for nonlinear transition dynamics. Two dierent analytical methods based on timedelayed feedback from previous literature have also been considered. Both the SPDFs https://doi.org/10.1088/1742-5468/aa50dc

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Figure 14. Eect of time delay τ, delay feedback intensities K1, K2 and noise intensity σ on the amplitude SPDF Qst(A). The parameter values are c = 0.01 (a) K1 = 0.9, K2 = 0.1, σ = 0.1; (b) τ = 0.9, K2 = 0.1, σ = 0.1; (c) τ = 0.9, K1 = 0.9, σ = 0.1; (d) τ = 0.9, K1 = 0.9, K2 = 0.1.


Figure 16. (a) Eect of damping coecient c on the amplitude SPDF Qst(A). (b) The mean A st as a function of damping coecient c. The parameter values are τ = 0.9, K1 = 0.9, K2 = 0.1, σ = 0.1.

of displacement and amplitude analysis are performed. To check the credibility of the approximate methods in the time-delayed QZS-VI dynamical system, let us compare the approximate theoretical results of figure 3 with figure 14. In figure 3, the structure of the displacement SPDF becomes narrow with an increase of time delay and displacement feedback intensity; that is, higher time delay and displacement feedback intensity lead to a lower response of the system. In addition, the structure of the displacement SPDF becomes wider as the value of the velocity feedback intensity increases; that is, a higher velocity feedback intensity leads to a stronger response of the system. Those results are consistent with the results in figure 14, which implies that the approximate https://doi.org/10.1088/1742-5468/aa50dc

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Figure 15. The mean amplitude A st as functions of (a) time delay τ, (b) delay feedback intensities K1, (c) K2 and (d) noise intensity σ, respectively. The parameter values are the same as in figure 14.


methods in the time-delayed QZS-VI dynamical system, combined with harmonic and Gaussian white noise excitations, are credible.

5. Concluding remarks

(i) The eective displacement potential is enhanced as the value of the time delay and displacement feedback intensity increases, but reduces as the value of the velocity feedback intensity and noise intensity increases. In all cases, it appears that the eects of control parameters on the eective displacement potential and the SPDF of displacement are consistent. (ii) The mean displacement increases with increasing time delay, displacement feedback intensity or harmonic excitation magnitude, but decreases with increasing velocity feedback intensity or noise intensity. (iii) For the noise-induced transition case, the stability of the equilibrium state xA is weakened by noise when there is weak displacement feedback intensity, strong velocity feedback intensity or a strong negative excitation magnitude, while the stability of the equilibrium state xA is enhanced by noise when there is strong displacement feedback intensity, weak velocity feedback intensity or a weak negative and positive excitation magnitude. (iv) For the time delay-induced transition case, the new kinds of RA and DES phenomena are investigated in our paper. The RA and DES eects can be strengthened by enhancing displacement or velocity feedback intensity, and can alsobe weakened by enhancing noise intensity.


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The main objective of this work is to extend the concept of nonlinear transition dynamics, that have been proven as useful and reliable for random perturbation, to a timedelayed QZS-VI dynamical system aected by combined harmonic and Gaussian white noise excitations. A physical explanation of the mechanisms for a noise-driven trans ition of a vibration isolator is derived from the eective potential of the FPK equation. We have shown that for a time-delayed QZS-VI dynamical system, a modified smooth and discontinuous oscillator, it is possible to formulate the problem in the Itô stochastic sense and present the FPK equation governing the evolution of the transition SPDF. Time delay, harmonic excitation magnitude, delay feedback and noise intensities have been taken into account in the control parameters. Due to the geometrical nonlinear properties, the QZS-VI dynamical system shows bistability in the deterministic case without time delay, i.e. there are two equilibrium states (xA and xB) separated by an unstable state. Our analysis results of the eective displacement potential, SPDF of displacement and the escape rate identify the occurrence of the phenomena of trans ition, RA and DES in the dynamics of the time-delayed QZS-VI system. The main observation includes the following points:


(v) A higher time delay, displacement feedback intensity and damping coecient lead to the lower response of the QZS-VI system, i.e. time delay and displacement feedback intensity can play the role of a damping force, while higher velocity feedback and noise intensities lead to a stronger response. (vi) Finally, we have checked that there is a good agreement between the delay nonlinear Langevin approach and the stochastic averaging method.

Acknowledgments The authors would like to acknowledge the financial support from the Natural Science Foundation of China (Grant No. 11372082 and 11572096) and the National Basic Research Program of China (Grant No. 2015CB057405). References [1] Lee Y Y et al 2009 The eect of modal energy transfer on the sound radiation and vibration of a curved panel: theory and experiment J. Sound Vib. 324 1003–15 [2] Al-Saif K A, Aldakkan K A and Foda M A 2011 Modified liquid column damper for vibration control of structures Int. J. Mech. Sci. 53 505–12 [3] Casciati F, Rodellar J and Yildirim U 2012 Active and semi-active control of structures theory and applications: a review of recent advances J. Intell. Mater. Syst. Struct. 23 1181–95 [4] Li H et al 2012 Reliable fuzzy control for active suspension systems with actuator delay and fault IEEE Trans. Fuzzy Syst. 20 342–57 [5] Yao W et al 2011 Design and analysis of the droop control method for parallel inverters considering the impact of the complex impedance on the power sharing IEEE Trans. Ind. Electron. 58 576–88 [6] Xue X et al 2011 Semi-active control strategy using genetic algorithm for seismically excited structure combined with MR damper J. Intell. Mater. Syst. Struct. 22 291–302 [7] Malek-Zavarei M and Jamshidi M 1987 Time-Delay Systems: Analysis, Optimization and Applications (New York: North-Holland) [8] Stépán G 1989 Retarded Dynamical Systems: Stability and Characteristic Functions (Essex: Longman Scientific and Technical) [9] Reddy D V R, Sen A and Johnston G L 1998 Time delay induced death in coupled limit cycle oscillators Phys. Rev. Lett. 80 5109 [10] Reddy D V R, Sen A and Johnston G L 2000 Experimental evidence of time-delay-induced death in coupled limit-cycle oscillators Phys. Rev. Lett. 85 3381 [11] Hu H Y and Wang Z H 2002 Dynamics of Controlled Mechanical Systems with Delayed Feedback (Berlin: Springer) [12] Zhang X X and Xu J 2015 Identification of time delay in nonlinear systems with delayed feedback control J. Franklin Inst. 352 2987–98 [13] Agrawal A K and Yang J N 1997 Eect of fixed time delay on stability and performance of actively controlled civil engineering structures Earthq. Eng. Struct. Dyn. 26 1169–85 [14] Hu H Y, Dowell E H and Virginv L N 1992 Resonances of a harmonically forced dung oscillator with time delay feedback control Nonlinear Dyn. 15 311–27 [15] Yu P, Yuan Y and Xu J 2002 Study of double Hopf bifurcation and chaos for an oscillator with time delayed feedback Commun. Nonlinear Sci. Numer. Simul. 7 69–91 [16] Xu J and Chung K W 2003 Eects of time delayed position feedback on a van der Pol-Dung oscillator Physica D 180 17–39 [17] Li X et al 2006 The response of a Dung–van der Pol oscillator under delayed feedback control J. Sound Vib. 291 644–55


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To improve the performance of the QZS isolator under combined harmonic and stochastic excitations, other types of nonlinear transition dynamics, e.g. stochastic bifurcation and resonance in the vibration isolation dynamics, should be investigated.



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