Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback

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Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback Chun-Xia Liu, Yan Yan, Wen-Quan Wang∗ Department of Engineering Mechanics, Kunming University of Science and Technology, Kunming, Yunnan, 650500, PR China

a r t i c l e

i n f o

Article history: Received 16 October 2018 Revised 10 October 2019 Accepted 16 October 2019 Available online xxx Keywords: MEMS/NEMS resonators Time delay Multiple scales method Melnikov function Chaos

a b s t r a c t The resonance and chaos of micro (nano) electro mechanical resonators with time delay feedback is concerned in the paper. Based on the experimental results, a lumped single degree-of-freedom (1DOF) model is studied and the effects of time delay displacement and velocity feedback on the system are investigated. In order to have a deep insight into the system, the amplitude frequency response curve of the system is firstly obtained using the multiple scales method. The Melnikov function method is then extended to the two time delay systems, and the analytically required condition for chaos was obtained. Finally, the fourth-order Runge–Kutta method, point-mapping method and spectrum diagram are used to simulate the evolution of the dynamic behavior of the time delay control system. Also, the stability of this time delay control system is studied thoroughly. The results show that time delay feedback is a good method for the control system and that reasonable selection of control system parameters can effectively suppress the vibration level for micro/nano-electro-mechanical resonator systems. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Extensive research on static and dynamic analyses of micro- and nano-electro-mechanical-systems (MEMS/NEMS) [1,2] under electrostatic actuation have been carried out because of their specific applications, such as MEMS based memory [3], switches [4], filters [5], microvalves [6], microrelays [7], and so on. Resonators are among the most commonly used components for many MEMS based systems and are beneficial in a variety of applications, such as sensors, accelerometers, and communication and signal processing devices. Delays in MEMS/NEMS devices are very common, which can be introduced into devices unavoidably or by design. For micro-electromechanical resonators, many inherent system delays can be introduced through actuators, processor dynamics, filters, and feedback measurements. In recent years, Duffing oscillator with delayed effects [8,9] have been studied. The analytic procedure and conclusions provide a reference for studying similar fractional-order dynamic systems with time delays. Then, Olgac et al. [10] introduced a dynamic vibration absorber with time delay position feedback in a single degree-offreedom system. The eigenvalue analysis method was used to determine the analytical solution of the feedback gain coefficient and time delay. Their research shows that the time delay dynamic vibration absorber can completely eliminate the vibration of the main structure. Olgac et al. [11] also designed a frequency-fixed linear time delay vibration absorber to control the vibration caused by two different frequency external excitations and assessed it using eigenvalue analysis. The authors found that the feedback gain coefficient and time delay completely eliminated the vibration of the main system, and they ∗

Corresponding author. E-mail address: [email protected] (W.-Q. Wang).

https://doi.org/10.1016/j.apm.2019.10.047 0307-904X/© 2019 Elsevier Inc. All rights reserved.

Please cite this article as: C.-X. Liu, Y. Yan and W.-Q. Wang, Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.047

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used elastic beam experiments to verify the damping control effect of frequency modulation (FM). In addition, Olgac and his collaborators [12] used time delay damping to perform fundamental research on torsional vibration. Their research verified the effectiveness of time delay damping in torsional vibration control. Xu et al. [13,14] performed the nonlinear research of time delay dynamics. In their work, the optimal value of the time delay was obtained based on the vibration dissipation time using eigenvalues analysis, and they established the relationship between the parameters and vibration properties of a nonlinear isolation system. In Ref. [15], the authors added a time delay nonlinear dynamic vibration absorber to a singledegree-of-freedom system and studied the damping performance of the main system using a multiple scales method. In the amplitude-delay response curve of the main system, based on the determined feedback gain coefficient, there was a time delay adjustment zone that allowed the main system to dampen the vibration. Hu et al. [16] used a multiple scales analysis method to discuss the main resonance, subharmonic resonance and stability of a class of weakly nonlinear time delay Duffing oscillators subjected to harmonic excitation. The authors analyzed the stability and determined the criteria for a single degree of freedom and multiple degrees of freedom, the law of stability switching, the infinite number of Hopf bifurcations, and the periodic vibration of their coexistence, and they answer many of the questions that could not be explained by the “modal overflow” problem. However, although much research has been performed on dynamic stability analysis of time delays, it is still difficult to analyze the stability of multi-delay dynamics and nonlinear time delay dynamics. Sato et al. [17] studied the influence of time delays on the free vibration and forced vibration of a nonlinear vibration system. Using the central manifold theorem and averaging method, the stable periodic solution of the time delay system and its stability were studied. Ji et al. [18] used the multiple scales method to analyze the primary resonant, superharmonic and subharmonic responses of a single-degree-of-freedom system under two different time delays. The results of the study verified the effectiveness of the double-delay effect on a nonlinear system. For an electrostatically actuated MEMS resonant sensor, Shang et al. [19,20] investigated the effectiveness of delayed position feedback in reducing the pull-in instability, and the effect on controlling the erosion of safe basins was shown. In another paper [21], Shang and Wen studied a MEMS resonator with delayed position feedback. They found that the delayed feedback control is effective on suppressing chaos of the micro mechanical resonator. Shao et al. [22] presented the effect of delayed velocity feedback on the response and stability of the system for small vibration amplitude and voltage loads, and showed that positive time delay can considerably strengthen the system stability. However, negative time delay feedback control gain can lead to unstable responses. Siewe [23] studied the effects of three different active controllers on the chaotic dynamical system and found that the vibration of the system can be controlled actively via negative velocity feedback. The concept of delayed resonator (DR) was proposed in the 1990s by the research group of Olgac [10,24]. It belongs to the class of active vibration absorbers where the main objective is to enhance the efficiency and flexibility in vibration suppression compared to the passive absorbers. The positive impact of passive absorbers to vibration amplitude reduction has been known for decades [25,26]. However, the frequency band where the passive absorber suppresses the vibration efficiently is relatively narrow, being centered at the natural frequency of the absorber. Besides, due to the fact that the absorber is never ideal, the vibration is not damped entirely even if the vibration frequency is identical with the natural frequency of the absorber. In order to improve and avoid the flaw of passive absorbers, DR approach has to be developed. An advantage of using DR feedback lies in the simplicity of the implementation and the easy tuning rules (a) removes oscillations of the primary structure for a very wide range of frequencies and (b) the tuning of the absorber is done in real time [10,24]. Recently, the analysis and design aspects of DR absorber with position, velocity or acceleration feedback have been widely applied in the literature [27–30]. Hence, in our paper, the nonlinear behaviors of micro-electromechanical DR is considered. The frequency response of the control system was determined using multiple scales analysis, then, the analytically required condition for chaos in this system are investigated using the Melnikov method. The effects of displacement and velocity feedback gain coefficients as well as their time delays on chaotic behavior are graphically demonstrated and discussed. Finally, the fourth-order Runge– Kutta method, point-mapping method and spectrum diagram are used to simulate the evolution of the dynamic behavior of the time delay control system. 2. Mathematical modeling In this section, a simplified model of a micro-electro-mechanical silicon resonator under combined alternate current (AC) and direct current (DC) actuation is shown in Fig. 1. The driving force on the resonator is achieved by means of an electric drive voltage which causes an electrostatic excitation between the electrode and the resonator. A microscope image of the clamped-clamped beam resonator can be seen in Fig. 2 [31]. It is worth noting that the gray material is silicon (Si), thin black line represents lithography etch gaps, and the white granular material corresponds to the aluminum (Al) key pad and the wire. The governing equation of motion is as follows according to Ref. [31]:

me f f z¨ + cz˙ + k1 z + k3 z3 =

1 1 Cv Cv V2 (VDC + VAC sin t )2 − 2 (d − z )2 2 (d + z )2 DC

(1)

where z, d, meff , k1 , k3 , c and  are the vertical displacement of the beam, initial gap width, effective lumped mass, linear mechanical stiffness, cubic mechanical stiffness, damping coefficient and frequency of AC voltage, respectively. Meanwhile, Cv , VDC and VAC represent the capacitance of the parallel-plate structure, bias DC voltage parameter and AC excitation voltage parameter, respectively. Suppose that the amplitude of the AC excitation voltage is much lower than the bias DC voltage. Please cite this article as: C.-X. Liu, Y. Yan and W.-Q. Wang, Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.047

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Fig. 1. Schematics of MEMS resonator.

Fig. 2. A microscope image of MEMS resonator.

Eq. (1) is from experiments of a clamped-clamped beam resonator and nonlinearities in MEMS silicon resonators are caused by different effects. Depending on the resonator layout, different nonlinearities may be dominant in the resonator response. Based on experimental results, Mestrom et al. [31] established a lumped single degree-of-freedom (1DOF) model and a Duffing-like expression (does not consider the effect of the quadratic and higher nonlinear terms) is adopted. With the proposed model, the observed behaviors are captured, and a quantitative match between the simulation and experimental results is established. The Duffing equation is a classical nonlinear dynamic equation with a spring force containing both a linear and a cubic term. It is well known that a Duffing model can describe the response of a periodically excited mass-spring-damper system and investigate the nonlinear effects like sub- and superharmonic resonances and softening and hardening behaviors [32–34]. It is worth noting that 1 DOF model is unable to represent an accurate result than the ones based on a distributed modeling approach. However, it captures essential nonlinear dynamics [35,36] and carries out a more straightforward nonlinear dynamic analysis. Eq. (2) is the non-dimensionless expression of Eq. (1). The forms of Eqs. (1,2) are widely used in refs. [37,38] for nonlinear dynamics analysis. The non-dimensionless expression of Eq. (1) can be expressed as follows [22,39]:



x¨ + μx˙ + α x + β x3 = γ

1

(1 − x )2

−

1

( 1 + x )2



+

A

(1 − x )2

sin (ωt )

(2)

Please cite this article as: C.-X. Liu, Y. Yan and W.-Q. Wang, Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.047

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Fig. 3. Schematic diagram of the controlled system [49].

The non-dimensional parameters in Eq. (2) are listed as follows:

x=

γ =

z , ωs = d



2 CvVDC

k1 k1 k3 d 2 , t = ωs t0 , α = ,β = , 2 me f f me f f ωs me f f ωs2

2me f f ωs2 d3

, A = 2γ

VAC  c ,ω = ,μ = . VDC ωs me f f ωs

where ωs is the purely elastic natural frequency. The paper focuses on the vibration control of MEMS resonator by a delayed feedback proportional-derivative (PD) controller. It should be pointed out that, compared with other controllers, PD controller has been widely used in giving global asymptotic stabilization of many systems, mostly due to their conceptual simplicity and explicit tuning procedures [40–42]. And it is well known that the derivative control is to enhance the stability [43,44]. Hence, considering fast dynamic response and short adjustment time in industrial applications, we integrate the time delayed displacement and velocity feedback into MEMS resonator model as shown in Fig. 3. In the control-loop, the motion state (displacement, velocity, acceleration, etc.) of the micro-electro-mechanical resonator measured by the sensor is transferred to the PD controller, and then the vibration of the system is controlled by the actuator. As a result, the modified dimensionless equation becomes the following:



x¨ + μx˙ + α x + β x = γ

1

3

(1 − x )2

−



1

+

(1 + x)2

A

(1 − x )2

sin (ωt ) + ux(t − τ1 ) + vx˙ (t − τ2 )

(3)

where u, v, τ 1 and τ 2 are the displacement feedback coefficient, velocity feedback coefficient, time delays of displacement and velocity feedbacks, respectively. To describe the system dynamics, the nonlinear electrostatic terms of Eq. (3) can be expanded based on Taylor’s expansion method which has been used in refs. [22,39,45]. The method leads to sufficient accuracy and will prove reasonable in Section 3. The expanded terms of the nonlinear terms are obtained as follows:





x¨ + μx˙ + c1 x + c3 x3 = A 1 + 2x + 3x2 + 4x3 sin (ωt ) + ux(t − τ1 ) + vx˙ (t − τ2 )

(4)

where c1 = α − 4γ , c3 = β − 8γ . 3. Analytical solution In this section, we use the multi scale method to obtain an accurate analytical solution of MEMS resonators. To determine the motion around x = x0 , where x0 is one of the equilibrium points [46,47], we introduce the following change of variables:

x(t ) = x0 + q(t )

(5)

Substitute Eq. (5) into Eq. (4) and the nonlinear terms in powers of q as follows:

q¨ + μq˙ + ω

2 0q

2

3



2

+ c2 q + c3 q = F 1 + k1 q + k2 q + k3 q

3





sin (ωt ) + u



c1 − + q(t − τ1 ) c3

+ vq˙ (t − τ2 )

(6)

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where

ω0 = F





√ c1 c1 c1 −2c1 , c2 = 3 −c1 c3 , c4 = 1 + 2 − − 3 + 4 c3 c3 c3

2 = Ac4 , k1 = c4







c1 c1 3 1+3 − −6 , k2 = c3 c3 c4





 −

5

c1 c3



c1 4 1+4 − , k3 = . c3 c4

Supposing a small perturbation parameter ε , q(t) can be replaced by q(t) = ε y(t). using the new transformations

μ = ε μ1 , F = ε 2 F1 , u = ε 2 u1 , v = ε 2 v1 Eq. (6) can be rewritten as follows:





y¨ +ε μ1 y˙ + ω02 y + c2 ε y2 + c3 ε 2 y3 = ε F1 1 + ε k1 y + ε 2 k2 y2 + ε 3 k3 y3 sin (ωt )



+ ε u1

−

(7)

c1 + ε 2 u1 y(t − τ1 ) + ε 2 v1 y˙ (t − τ2 ) c3

These assumptions are valid for a MEMS resonator with a small AC amplitude and large quality factor with respect to the bias voltage. If the frequency of the actuation is close to the fundamental frequency, we have

ω = ω 0 + σ ε 2 = ω 0 + ω

(8)

Where σ is the nonlinear detuning parameter. Let Tn = ε n t,(n = 0, 1, 2) and the displacement be y0 (t) = y0 (T0 , T1 , T2 ). Eq. (7) is expanded as follows:

y(t ) = y0 (T0 , T1 , T2 ) + ε y1 (T0 , T1 , T2 ) + ε 2 y2 (T0 , T1 , T2 ) · ··

(9)

yτi (t ) = y0τi (T0 , T1 , T2 ) + ε y1τi (T0 , T1 , T2 ) + ε 2 y2τi (T0 , T1 , T2 ) · ··, i = 1, 2

(10)

The time derivatives are defined as follows:

d = D0 + ε D1 + ε 2 D2 + · · · dt

(11)

  d2 = D0 2 + 2 ε D0 D1 + ε 2 D1 2 + 2 D0 D2 + · · · dt 2

(12)

By substituting Eqs. (9-12) into Eq. (7) and equating the coefficients of ε , we have

ε 0 : D20 y0 + ω02 y0 = 0



ε 1 : D20 y1 + ω02 y1 = −μ1 D0 y0 − 2D0 D1 y0 − c2 y20 + u1 −

(13) c1 + F1 sin (ωt ) c3

(14)

  ε 2 : D20 y2 +ω02 y2 = −2D0 D1 y1 − D21 + 2D0 D2 y0 − μ1 D1 y0 − μ1 D0 y1 − 2c2 y0 y1 − c3 y30 + F1 k1 y0 sin (ωt ) + u1 y0τ1 + v1 D0 y0τ2

(15)

The general solution of Eq. (13) is

y0 = Z (T1 , T2 )eiω0 T0 + Z¯ (T1 , T2 )e−iω0 T0

(16)

where Z and Z¯ are the complex amplitude and complex conjugate of Z, respectively. Substitute Eq. (16) into Eq. (14), and then,



D20 y1 +ω02 y1 = −



μ1 i ω 0 Z + 2 i ω 0 D 1 Z +



− c2 Z 2 e2iω0 T0 + 2c2 Z Z¯ + c2 Z¯ 2 e

iF1 iσ T1 iω0 T0 e e − 2

 −2iω T 0 0



+ u1

To eliminate the secular terms, we have

y1 =

 c2  2 2iω0 T0 Z e + Z¯ 2 e−2iω0 T0 + 2Z Z¯ + u1 2 3ω0

D1 Z = −

μ1 2

Z−

F1 iσ T1 e 4ω0

 −

c1 − c3

c1 c3

  iF μ1 iω0 Z + 2iω0 D1 Z + 1 e−iσ T1 e−iω0 T0 2

(17)

(18)

(19)

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Substituting (16), (18), and (19) into (15) and eliminating the secular terms gives

ic22 2 iμ1 F1 iσ T1 iμ21 Z ic2 u 1 D2 Z = − e − + Z Z¯ + 8ω0 ω0 16ω02 ω02 iv1 Z v1 Z + cos (ω0 τ2 ) − sin (ω0 τ2 ) 2 2 Assuming Z =

aeiθ 2



−

c1 iu 1 Z u1 Z 3 ic3 2 Z+ Z Z¯ − cos (ω0 τ1 ) − sin (ω0 τ1 ) c3 2ω0 2ω0 2ω0

(20)

, substituting it into Eqs. (18–20), and separating the real and imaginary parts, we have

μa

ua ε F1 εμF1 va a˙ = − − cos (ϕ ) + sin (ϕ ) − sin (ω0 τ1 ) + cos (ω0 τ2 ) 2 2ω0 8ω0 2ω0 2  c1 3 ε 2 c3 a3 ε F1 εμF1 μ2 a ε 2 c22 a3 c2 ua au aϕ˙ = aσ + sin (ϕ )+ cos (ϕ ) + − − − − + cos (ω0 τ1 ) 2ω0 8ω0 8ω0 4ω0 ω0 c3 8ω0 2ω0 av + sin (ω0 τ2 )

(21)

(22)

2

where σ T1 − θ = φ . Letting a˙ = aϕ˙ = 0, we obtain the frequency response equation

F2 4ε 2 ω

2 0

+

 μa μa 2 μ2 F 2 va = + sin ω τ − cos ω τ ( ) ( ) 0 1 0 2 2 2 2ω0 2 64ε 2 ω0   2 2 2 3 2 c2 ua c1 3 ε 2 c3 a3 μ a ε c2 a au av + aσ + − − − − + cos (ω0 τ1 ) + sin (ω0 τ2 ) 8ω0 4ω0 ω0 c3 8ω0 2ω0 2

(23)

Via Eq. (23), the peak value of the primary resonance can be expressed as

amax =

εμF1 8 ω 0 μ0

(24)

μ+ ωu sin ω0 τ1 −v cos ω0 τ2

0 where μ0 = . 2 To uncontrolled system, the corresponding peak amplitude can be expressed as

a¯ max =

ε F1 4ω0

(25)

As we all know, it is very difficult to solve the analytical expression of a nonlinear vibration system. Thus, an attenuation ratio R is defined according to Refs. [48,49]

R=

amax = a¯ max

1 u ω0 sin ω0 τ1 −v cos ω0 τ2

1+

.

(26)

μ

then the performance of vibrating control is evaluated directly using Eq. (26). In order to simplify the analysis, the delays are expressed as τ 1 = τ and τ 2 = ϕ + τ . According to the suggestion of Ref. [18], the phase difference ϕ can be assumed as π2 because the phase displacement is delay of velocity by π2 . So Eq. (26) can be rewritten as

R=



1+

1 u ω0



+v sin ω0 τ

(27)

μ

The eigenvalues of the corresponding Jacobian matrix of Eqs. (21) and (22) relate with the system stability. The calculation process is detailed in Appendix A. The eigenvalue can be obtained from the following equation

   λ2 + 2μ0 λ + μ20 + σ0 − v0 a20 σ0 − 3v0 a20 = 0

(28)

The sufficient condition of system stability can be expressed as

f (σ0 ) = μ20 +



  σ0 − v0 a20 σ0 − 3v0 a20 > 0

(29)

or the critical equation f(σ 0 ) = 0 has no real solution. Then inequality (29) can be written as

μ20 ≥ v20 a4max

(30)

Substituting Eq. (24) into inequality (30), we obtained



gτ 1 sin ω0 τ

1 ≥ 2

3

v0 μ2 ε 2 F12 − μ, μ0 > 0 ω02

(31)

This is the scope of the stable vibration control parameters of system. Please cite this article as: C.-X. Liu, Y. Yan and W.-Q. Wang, Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.047

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7

Fig. 4. Amplitude-frequency curve of the system with different delays and feedback gain coefficients.

If f(σ 0 ) = 0 has two real solutions, we obtain the scope is

  2 ε 4 F14 ε 2 c22 3 2 ε 2 F12  2 2 3 2 c1 μ2 2 c2 u 2 − 2σ − ε gτ 2 cos ω0 τ ≥ c + ε c + 2 + ε c − μ + − , μ0 > 0 3 3 2 2 4 8 4ω0 ω0 c3 64ω03 84 ω06

or

(32)



  2 ε 4 F14 ε 2 c22 3ε 2 c3 c1 μ2 2 c2 u 2 gτ 3 cos ω0 τ ≤ −2 + − μ − 2σ − + − , μ0 > 0 6 4 4 8 4 ω ω c 8 ω0 0 0 3

(33)

where gτ i = ( ωu + v ), i = 1, 2, 3. 0

So far, we can conclude that for the critical equation, the optimal control parameters meets min ity (31) when it has no actual solution. The optimal control parameters satisfy min 1+

min 1+

1 ( ωu +v) sin ω0 τ 0 μ

1 ( ωu +v) sin ω0 τ 0 μ

1 g sin τ 1+ τ 1μ

and inequal-

and inequality (32), or

and inequality (33) if it has two real solutions.

To investigate vibration amplitudes of DR before and after control, the selected system parameters are: α = 1, β = 12 and

γ = 0.338 [46,47].

In Fig. 4(a) and (b), the Amplitude-frequency curves of the system by the multiple time scales method and the fourthorder Runge–Kutta have been simulated. Obviously, the figure shows an excellent agreement between two methods. Next, to clear how the amplitude varies with the effects of the feedback gain coefficients u, v and delay feedbacks τ 1 ,τ 2 , the Amplitude-frequency curves for the primary resonance case by multi scale method are plotted in Fig. 5. The implementation in the figure represents stability and the dashed line indicates instability. In view of the periodicity of time delay system, the discussion in the following is in one period. Fig. 5(a) illustrates the effect of the amplitude of the AC actuation voltages on the amplitude of the steady-state response of the system without a time delay. With increase of A, the response area broadens and the maximum amplitude of vibration rises up, thus demonstrating that the increase of the applied load amplitude increases the amplitude of the nonlinear resonance. It is inferred that increasing A also extends the bandwidth of the resonance zone frequency. The influence of the viscous damping coefficient on the response curve is shown in Fig. 5(b). As the damping coefficient raise, the peak becomes smaller, the migratory quantity of the resonance frequency decreases, the instability is reduced, and the non-linearity is weakened. The curve of the characteristics of hard springs and multi-valued areas are significantly reduced, which is agreement with the results of Ref. [14]. It is observed in Fig. 5(c) that when the gain coefficient of displacement feedback raises, although the peak shifts to the right, the change of the peak is not obvious and the response spring characteristic and nonlinearity do not change. However, as shown in Fig. 5(d), when v = 0.3 and 0.5, the peak amplitude is obviously decreased with the increase of the feedback control gain. It is also illustrated in Fig. 5(e), the increase of u only let the peak shift to a high frequency, however, the increase of v can effectively suppress the vibration amplitude. Therefore, optimizing the velocity and displacement feedback gain coefficient can effectively control the vibration of a nonlinear system. Influence of time delay on the frequency response of a MEMS resonator is shown in Figs. 5(f)–(h). From Figs. 5(f) and (g), it is observed that both the displacement time delay τ 1 and velocity time delay τ 2 can suppress the vibration amplitude. The effectiveness of the time delay is further investigated in Fig. 5(h). Four cases including without time delay, only displacement delay, only velocity delay and both two time delays. It is concluded that velocity delay has a more obvious effect comparing with displacement delay and the two-time delays have a better suppression on system vibration. The response Please cite this article as: C.-X. Liu, Y. Yan and W.-Q. Wang, Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.047

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Fig. 5. Amplitude-frequency curve of the system with different delays and feedback gain coefficients.

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Fig. 5. Continued

curves of the primary resonance with both the feedback gain coefficient and time delay are shown in Fig. 5(i). Obviously, with the increase of the time delay (within one periodic), there was a time delay adjustment zone that allowed the main system to dampen the vibration, the nonlinear suppression effect of the system may be enhanced. Results in Fig. 5(f–i) further suggest that time delayed system may lose its stability at some values of time delays. However, time delays may improve the stability of the system elsewhere. The conclusions accord well with those in Ref. [50]. The relations between the attenuation ratio and time delay with different feedback gain coefficients are presented in Fig. 6. The figure suggests that reasonable selection of time delays for different feedback gain coefficient obtains a small value of the attenuation rate R. The smaller R is, the better vibration control of system is. Obviously, π2 is one of the optimal time delay.

4. Chaos prediction For a delay feedback system, the Melnikov function method can still be extended to a time delay system to determine the necessary conditions for global bifurcation of the system. The precondition ensures that the occurrence of the time feedback will not change the stability of the equilibrium point of the original system. That is, the stability of the equilibrium point does not change as the control parameters vary in the system. Next, to use the Melnikov method, we make changes as follows:

x˙ = y, μ = ε μ1 , A = ε A1 , u = ε u2 , v = ε v2 Please cite this article as: C.-X. Liu, Y. Yan and W.-Q. Wang, Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.047

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Fig. 6. Variation of R with time delay for different feedback gain coefficient at μ = 0.06.

Eq. (3) becomes

x˙ = y



 

y˙ = −α x − β x3 + γ 4x + 8x3 + o x5

+ ε −μ1 y +



A1

(1 − x )2

sin (ωt ) + u2 x(t − τ1 ) + v2 y(t − τ2 )

(34)

When τ = 0, the Melnikov function of (34) is identical with that described by Hossein et al. [39]. According to the Hopf bifurcation theorem, the critical condition of Hopf bifurcation in Eq. (3) is as follows:

( 3β − 8γ ) ( 1 − 4γ ) − u cos (ω0 τ1 ) = 0 ( β − 8γ ) μω0 − ω0 v cos (ω0 τ2 ) + u sin (ω0 τ1 ) = 0

−ω02 − ω0 v sin (ω0 τ2 ) + α − 4γ −

(35)

Next, we consider the critical values of different time delay. When only the delayed displacement feedback exists, Eq. (34) become

−ω02 + α − 4γ −

( 3β − 8γ ) ( 1 − 4γ ) = u cos (ω0 τ1 ) − μω0 = u sin (ω0 τ1 ) ( β − 8γ )

Suppose I1 is a set of (μ, u), then

I1 =

⎧ ⎨ ⎩

2α − 8γ −

2 ( 3β − 8γ ) ( 1 − 4γ ) − μ2 > 0 β − 8γ

(μ, u )|  μ4 − 4 μ2 α − 4 γ −



(3β −8γ )(1−4γ ) β −8γ

(36)

⎫ ⎬ (37)

⎭

+ 4u > 0 2

When (μ, u) ∈ I1 , (36) has two unequal positive roots ω0 + > ω0 − > 0. According to Hopf bifurcation theorem, the critical value of the displacement time delay for Hopf bifurcation is

τ1+ =

1





2π − arccos

ω0+

2 −ω0+ + α − 4γ

u

( 3β − 8γ ) ( 1 − 4γ ) − u ( β − 8γ )



(38)

Note that the stability of the equilibrium point of the system (34) does not change when 0 ≤ τ 1 < τ 1 + . Similarly, the critical value of the velocity time delay for Hopf bifurcation is

τ2+ =

1



ω0+

2π − arccos

 μ

(39)

v

Obviously, when 0 ≤ τ 2 < τ 2 + , the stability of the equilibrium point of the system (34) does not change. For a weak feedback and small delay, the delay displacement and velocity feedback terms can be regarded as perturbation terms. Therefore, the corresponding Melnikov function of Eq. (34) is

M± (t0 ) = ε



+∞ −∞



y± 1

(t ) −μ



± 1 y1

(t ) + 

A1

2 sin (ω (t + t0 ) ) +

1 − x± t 1( )

u2 x± 1

(t − τ1 ) + v

± 2 y1

(t − τ2 ) dt

(40)

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Fig. 7. Bifurcation diagram of amplitude with A.

Fig. 8. (a) Phase map, (b) Poincare map and (c) Frequency spectrogram of Eq. (2) when A = 0.0305.

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Fig. 9. Bifurcation diagram of amplitude with u.

Fig. 10. (a) Phase map, (b) Poincare map and (c) Frequency spectrogram of Eq. (3) when u = 0.25 and τ = 0.

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Fig. 11. (a) Phase map, (b) Poincare map and (c) Frequency spectrogram of Eq. (3) when u = 0.31 and τ = 0.

where q = (x1 , y1 ) is the solution corresponding to an unperturbed homoclinic orbit [39], which is introduced as follows [39]:

√  x± 1 (t ) = ±x sech α1 t ,

where x =



1 2

− βα −

√

 y± 1 (t ) = ∓x

4α 2 +4αβ +β 2 −32βγ 2β

√

√

√

α1 sech α1t × tanh α1t

(41)

and α 1 = 4γ − α .

By substituting (41) into (40) and using Mathematica, Melnikov’s function is defined as follows:

M (t0 ) = M1 + M2 + M3 + M4 Note that the values of M1 , M2 , M3 and M4 are obtained by Cauchy’s residue theorem, which can be expressed by the following:

M1 = −

2   2 √ x μ α1 3

   ω0 − arccos x M2 = ∓  cos (ω0 t0 ) sinh (ω0 π ) 1 − (x )2   2 √ √ √  M3 = −2 x ucsch α1 τ1 α1 τ1 coth α1 τ1 − 1 2x π Aω0

 2

M4 = 2 x where ω0 =

√

vα1 α1 csch3

sinh



√   √  √   √ α1 τ2 2 sinh 2 α1 τ2 − α1 τ2 3 + cosh α1 τ2

√ω . −α1

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Fig. 12. Bifurcation diagram of amplitude with τ 1 for u = 0.25.

Fig. 13. (a) Phase map, (b) Poincare map and (c) Frequency spectrogram of Eq. (3) when τ 1 = 0.22 and u = 0.25.

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Fig. 14. (a) Phase map, (b) Poincare map and (c) Frequency spectrogram of Eq. (3) when τ 1 = 0.56 and u = 0.25.

Fig. 15. Bifurcation diagram of amplitude with v.

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Fig. 16. (a) Phase map, (b) Poincare map and (c) Frequency spectrogram of Eq. (3) when v = 0.025 and τ 2 =0.

Let |M2 | > |M1 + M3 + M4 |, then

   sinh ω − arccos x   0      sinh (ω0 π )  )2   1− x (        2  √  √ − x μ α1 − 2x ucsch α1 τ1 √α1 τ1 coth √α1 τ1 − 1   >  3         √ √ √ √ √ + 2x vα1 α1 csch3 α1 τ2 2 sinh 2 α1 τ2 − α1 τ2 3 + cosh α1 τ2  2π Aω0

(42)

inequality (42) is the necessary condition of chaos. When only the delayed displacement feedback exists, inequality (42) becomes

   sinh ω − arccos x    √ √ √   0   2  √ >  x μ α1 + 2x ucsch α1 τ1 α1 τ1 coth α1 τ1 − 1     sinh (ω0 π ) 3  1 − (x )2  2π Aω0

(43)

When only the delayed velocity feedback exists, inequality (42) becomes

   sinh ω − arccos x   0      sinh (ω0 π )  )2   1− x (    2 x μ√α1  3   > √ √ √ √ 3 √ −2x vα1 α1 csch ( α1 τ2 ){2 sinh (2 α1 τ2 ) − α1 τ2 [3 + cosh ( α1 τ2 )]} 2π Aω0

(44)

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Fig. 17. (a) Phase map, (b) Poincare map and (c) Frequency spectrogram of Eq. (3) when v = 0.05 and τ 2 = 0.

Fig. 18. Bifurcation diagram of amplitude with τ 2 .

5. Numerical simulations In this section, the fourth-order Runge–Kutta method, point-mapping method and spectrum diagram are used to simulate the evolution of the dynamic behavior of the time delay control system. Assuming α = 1, γ = 0.338 [39] and β = 15, μ = 0.12, ω = 0.8. The bifurcation of the control system with the change of parameter A is shown in Fig. 4. Obviously, with the increase of A, the vibration of the Eq. (2) goes from simple periodic motion to double periodic bifurcation and, finally to chaos. When Please cite this article as: C.-X. Liu, Y. Yan and W.-Q. Wang, Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.047

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Fig. 19. (a) Phase map, (b) Poincare map and (c) Frequency spectrogram of Eq. (3) when v = 0.253 and τ 2 = 0.006.

A=0.0305 from Fig. 7, we observed that Eq. (2) is in a chaotic state, which can be verified by the random scatter point set in the Poincare cross-section and continuous spectrum in the power spectrum shown in Figs. 8(a)–(c).

5.1. Displacement feedback Firstly, the dynamical behaviors of the system with only the displacement feedback coefficient u existing is studied. The bifurcation diagram of u is shown in Fig. 9. When u = 0.0204, the system is in a chaotic state, which is consistent with the theoretical prediction by inequality (43), and find that the system is transformed from the initial chaotic motion to periodic motion with the increase of u. Furthermore, the values of u were chosen to be 0.25 and 0.31, respectively. The phase diagrams, Poincare cross-sections and spectrograms were simultaneously used to verify the validity of the conclusions. When u = 0.25, as observed in Figs. 10(a)–(c), the phase diagram is not repeated and disorganized. The irregularly scattered point set of the Poincare cross-section and the continuum of the spectrogram show that the system is in chaos. However, when u = 0.31, as shown in Figs. 11(a)–(c), the phase diagram is a simple closed curve and there are only three scattered points in the Poincare cross-section. There are only two distinct peaks in the spectrum. These phenomena imply that the system is in periodic motion. Obviously, choosing an appropriate displacement feedback coefficient can effectively control the system. Secondly, the dynamical behaviors of the system with the time delay τ 1 is studied for fixed u. The cases with u= 0.25 are plotted in Figs. 12–14. τ 1 + = 3.4840 is obtained from Eq. (38). Thus, the range of displacement time delay can be taken as [0, 1] ∈ [0, τ 1 + ). The bifurcation diagram of τ 1 is shown in Fig. 12. It can be easily seen from Fig. 12 that with the increase of τ 1 , the system changes from chaos to a stable periodic motion. The numerical simulation value of a critical chaotic state approximates 0.417 which is in agreement with that obtained by inequality (43). To further describe the evolution of system motion, numerical simulations are given in Figs. 13 and 14. Please cite this article as: C.-X. Liu, Y. Yan and W.-Q. Wang, Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.047

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Fig. 20. (a) Phase map, (b) Poincare map and (c) Frequency spectrogram of Eq. (3) when v = 0.253 and τ 2 = 0.23.

Figs. 13(a)–(c) show the simulation results of the system when τ 1 = 0.22. From the figure, we can see that the disorganized phase plane, the Poincare section’s irregularly scattered point set, and the spectrum’s continuous spectrum which indicates that the system is in chaos at this time. Figs. 14(a)–(c) show that when τ 1 = 0.56, the system is in periodic motion. Obviously, the phase plan has two closed curves at this time, the Poincare cross-section only shows two points, and the power spectrogram contains two peaks. Therefore, the appropriate value of the displacement feedbacks can help to control the movement of the system. 5.2. Velocity feedback Firstly, the dynamical behaviors of the system is studied as the velocity feedback coefficient v changes. To analyze the control effect of v on Eq. (3), the parameters used in the calculation are selected as α = 1, β = 12, γ = 0.338, μ = 0.03 [46–47] and ω = 0.08. Then, we show the bifurcation diagram of v in Fig. 15. From Fig. 15, we know that the system eventually moves towards a period. We select v = 0.025 and v = 0.05 in Figs. 16(a)–(c) and 17(a)–(c) and observe the phase diagram, Poincare crosssection, and frequency spectrum. Obviously when v = 0.025, the system is in chaos status. When v = 0.05, the system is in period motion. For another, the dynamical behaviors of the system is studied as the time delay τ 2 changes. The cases with ν = 0.253 are plotted in Figs. 18–20. According to Eq. (39), τ 2 + = 2.7813 is obtained. Thus, the range of velocity time delay τ 2 ∈ [0, 0.3] is reasonable in Fig. 18. Moreover, it is observed from the bifurcation diagram that the numerical simulation value of a critical chaos approximates 0.11 which accords well with that in inequality (44). With the increase of τ 2 , the controllable system enters periodic motion from chaotic motion. Furthermore, Fig. 19(a)–(c) and 20 (a)–(c) correspond to the phase diagram, Poincare cross-section, and frequency spectrum of the controlled system. In Fig. 19, when τ 2 = 0.006, the phase diagram is not repeated. Both the set of irregular points in the Poincare section and continuum and the different peaks of Please cite this article as: C.-X. Liu, Y. Yan and W.-Q. Wang, Resonance and chaos of micro and nano electro mechanical resonators with time delay feedback, Applied Mathematical Modelling, https://doi.org/10.1016/j.apm.2019.10.047

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the spectrogram indicate that the system is in chaos. However, when τ 2 = 0.23, as observed in Fig. 20, although the phase diagram is less obvious, the three points in the Poincare cross-section and two distinct peaks in the spectrogram indicate that the controlled system is in periodic motion at this time. From the above analysis, it is known that an appropriate time delay velocity feedback value, to a certain extent, has a certain control effect on the system. 6. Conclusions In the paper, a proportional-derivative (PD) controller is carried out to suppress the nonlinear vibrations of an electrostatically actuated clamped-clamped silicon beam resonator under DC and AC voltages. Based on the above discussion, some conclusions are as following: (1) Velocity feedback gain coefficient and two time delays are significant to improve the level of vibration control, while the displacement feedback gain coefficient is insensitive to mitigate the system vibration of micro-electro-mechanical resonator systems. Moreover, the two-time delays have a better suppressive effect on the system. (2) Time delayed system may lose its stability at some values of time delays. However, time delays may improve the stability of the system elsewhere. (3) The numerical simulation result of a critical chaotic state approximates to the theoretical prediction value by the Melnikov function method, which verifies the effectiveness for chaos control system. (4) The increase of the AC actuation voltage can cause chaos in the system. Yet, the displacement feedback coefficient, velocity feedback coefficient and their time delays can effectively restrain the chaotic motion of the system to a simple motion. Acknowledgments The work described in this paper is funded by the research Grant from the Natural Science Foundation of China (Grant nos. 11662006, 11172115). The authors are grateful for their financial support. Appendix A The stability of stead -state motion is determined by the properties of singular points in Eqs. (21) and (22). The following relationships are used:

a = a0 + a1 , ϕ = ϕ0 + ϕ1

(A1)

where a0 , φ 0 are the solution of Eqs. (21) and (22), a1 , φ 1 are the small parameters. Substituting Eq. (A1) into Eq. (21-22) and expanding for small perturbation, we derive

a˙ 1 = −μ0 a1 +

ϕ˙ 1 =

σ

0

a0

 εF

1

2ω0



sin ϕ0

− 3v0 a0 a1 +



ϕ1 +

 εμF

1

8ω0

cos ϕ0



ϕ1

 εF  εμF 1 1 cos ϕ0 ϕ1 − sin ϕ0 ϕ1 2 ω0 a0 8 ω0 a0

(A2) (A3)

Equation of the eigenvalues corresponding to the Jacobian matrix of Eq. (A2) and (A3) may be expressed as

  −μ0 − λ  σ0  a − 3 v0 a0 0

ε 2 c2

where v0 = 4ω 2 + 0

 σ0 a0 − v0 a30  =0 −μ0 − λ



−

3ε 2 c3 8ω0 .

(A4)

Then Eq. (28) is obtained.

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