Stability and perturbation analysis of a one-degree-of-freedom doubly clamped microresonator with delayed velocity feedback control

Abstract

In this paper, a study on a doubly clamped microresonator actuated by two symmetrical electrodes is carried out to investigate its dynamic properties with delayed velocity feedback control. A stability chart of the linearized system depicting delay time versus feedback gain is drawn first, which is actually nonperiodic. Moreover, stability switches do exist in this system. Then, the method of multiple scales is used to determine the existence, stability and dynamic properties of small amplitude vibration in the neighborhood of different equilibrium positions. It is shown that the stability condition via perturbation analysis overestimates the system stable region. The delayed stability condition via linearized analysis is more suitable for stability estimation. The following analytical and numerical results are presented to investigate frequency responses and frequency/damping trimming properties with various system and control parameters. Moreover, explicit formulas for optimum direct current (DC) voltage and equivalent natural frequency, corresponding to an approximate linear-like state, are deduced, respectively. Two typical design sketches depicting the initial gap width versus DC voltage are drawn with different beam lengths and thicknesses. Finally, a case study is carried out to verify the correctness of our analytical results about linear-like state prediction and frequency/damping trimming.

Keywords

Micro-electro-mechanical systems delayed velocity feedback stability nonlinear analysis multiple scales

1. Introduction

Recently, micro-electro-mechanical systems (MEMSs) have been developed for various applications in the field of sensing and actuating (Batra et al., 2007; Nayfeh et al., 2010; Younis, 2011; Zhang et al., 2014). As one core micro-component of the MEMS community, electrically actuated microbeams have attracted much attention and have been investigated by many scholars. Their concentrations mainly focus on the static or dynamic properties of microbeams, such as pull-in instability (Nayfeh et al., 2007; Krylov et al, 2008, Fang and Li, 2013), nonlinear characteristics (Luo and Wang, 2002; Haghighi and Markazi, 2010; Towfighian et al., 2011; Zaitsev et al., 2012; Chen et al., 2013; Younesian et al., 2014; Han et al., 2015a), control strategies (Park et al., 2008; Haghighi and Markazi, 2010; Lakrad and Belhaq, 2010; Song and Sun, 2013; Rahim and Younis, 2016) and some design analyses (Hu et al., 2004; Kouravand, 2011; Seleim et al., 2012; Han et al., 2015b). In this paper, based on our previous works (Han et al., 2015a, 2015b), an extended study on a doubly clamped microresonator actuated by two symmetrical electrodes (Haghighi and Markazi, 2010) is carried out to investigate its stability and dynamic properties with delayed velocity feedback control.

The delayed feedback signal can be displacement, velocity and acceleration (Shao et al., 2013). Generally, feedback controllers have the ability to stabilize the system response and enhance resonator performance (Hu and Wang, 2002). For instance, Ge and Lin (2003) used delayed displacement feedback control to effectively suppress the chaos of an electro-mechanical gyrostat system. Yamasue and Hikihara (2006) presented a delayed displacement feedback controller to stabilize the chaotic motion of micro-cantilevers in dynamic force microscopy and showed the experimental stabilization results in their subsequent study (Yamasue et al., 2009). Alsaleem and Younis (2010, 2011) used a delayed velocity/displacement feedback controller to effectively enhance the stability of microresonators driven electrostatically in a nonlinear regime and prevent them from pull-in instability. Rezaie and Jahed Motlagh (2011) addressed a new adaptive delayed displacement feedback controller to stabilize a class of chaotic time-delayed systems with a variable parameter. Nayfeh and Nayfeh (2012) proposed a delayed acceleration feedback control strategy to mitigate regenerative chatter in turning on lathes. Recently, by the application of the finite difference technique combined with Floquet theory, Masri et al. (2015) investigated the stabilization effect of a delayed velocity feedback controller for MEMS resonators undergoing large motion.

However, delayed feedback controllers with inappropriate control parameters may also deteriorate the control performance or even lead to unstable responses of systems (Hu and Wang, 1998; Hu et al., 1998). Thus, the effects of delayed feedback control on system performances still need to be considered. Wang et al. (2004b) studied the local dynamics of a one-degree-of-freedom (1-DOF) system with delayed velocity feedback and pointed out that feedback control could induce Hopf bifurcation. Global dynamics of a Duffing oscillator with delayed displacement feedback were also studied by Wang et al. (2004a). Stability switches and local bifurcations were presented and analyzed in detail. Sun et al. (2006, 2007) investigated the control mechanism of delayed displacement and velocity feedback control on a nonautonomous Duffing oscillator. The results showed that positive feedback could reduce the possibility of chaos, while negative feedback could enlarge the possible chaotic domain. El-Bassiouny (2006) presented an analysis of primary and subharmonic resonances of a cantilever beam under delayed displacement and velocity feedback control. Using the method of multiple scales (MMS), the effect of time delay was investigated and the equivalent damping concept related to delay feedback was proposed in his study. Xu et al. (2007) proposed a perturbation incremental scheme to study the delayed displacement feedback control on double Hopf bifurcation and nonlinear dynamics of a general dynamic system. Naik and Singru (2011) presented the resonance, stability and chaotic analyses of a quarter-car vehicle system with delayed displacement and velocity feedback, and pointed out the importance of feedback control on system performance, which should be taken into account for designing vehicle suspension systems. Siewe et al. (2012) presented a study on the Rayleigh–Duffing oscillator with delayed displacement and velocity feedback signals and found that appropriate choice of time delay could broaden the stable region of the nontrivial steady-state solutions and enhance control performance. Shao et al. (2013) used the MMS to investigate the small vibration of a microresonator with delayed velocity feedback control, and pointed out that positive feedback control could strengthen system stability, while negative feedback could lead to an unstable response.

From the above analyses, one can see that delayed feedback controllers are very important to the stability and response of dynamic systems. Analytical investigations are required to reveal the control mechanism and obtain appropriate control regions. In Shao et al. (2013), the effect of delayed velocity feedback control on a microresonator actuated by one electrode was investigated in detail. The function of the controller on system damping trimming was discussed analytically and numerically. Their work sheds light on the theoretical analysis of dynamic a MEMS with delayed feedback control. However, they only considered a fixed delay time (half period of free oscillation) and then realized system damping control. Detailed operation mechanisms of this type of controller on the microresonator with other delay times are still unclear. Furthermore, our early studies find that the stable region is actually nonperiodic and stability switches do exist in this system, which is different from the periodic stability conclusion described by Shao et al. (2013). Moreover, with the decrease of characteristic size in MEMSs, it seems to be feasible to obtain velocity signals of the microbeam through an extra electrode (Mestrom et al., 2008, 2009). All of the above motivate our present work.

The rest of this paper is organized as follows. In Section 2, an analytical model with delayed velocity feedback control is briefly introduced. The equilibrium positions of its Hamiltonian system are given based on our previous works (Han et al., 2015a, 2015b). In Section 3, the effect of delayed feedback control on system stability is carried out via linearization analysis. A stability chart induced by the delayed velocity feedback controller is investigated in detail. Then, the MMS is applied to determine the response and stability of small vibration around different equilibrium positions. In Section 4, analytical and numerical results are given to investigate the frequency responses and frequency/damping trimming properties with different system and control parameters. Some design tips on linear-like state and frequency/damping trimming are discussed and then numerically verified. Finally, the last section presents the discussion and conclusions.

2. Model description

A simplified 1-DOF model to describe an electrostatically actuated microresonator (Haghighi and Markazi, 2010) is shown in Figure 1, in which $\hat{x}$ is the vertical displacement and d₀ is the initial gap width. The resonator is deflected by direct current (DC) voltage $V_{dc}$ and actuated by alternating current (AC) harmonic voltage $V_{ac} \sin (Ω t)$ . Variation of the gap between the movable microbeam and the fixed electrode at the bottom can generate output voltage signals $V_{out}$ . Through signal acquisition and transformation (Mestrom et al., 2008, 2009), one can obtain the relationship between $V_{out}$ and velocity $d \hat{x} / d t$ . Then, the delay signal $d {\hat{x}}_{d} / d t$ can be generated by a stack of shift registers (Alsaleem and Younis, 2010, 2011). As our main focus is on the theoretical investigation of this system with delayed velocity feedback control, detailed implementation is not discussed here any further.

Figure 1.

Schematic of an electrostatically actuated microresonator with delayed velocity feedback control.

In fact, the actuation voltage and feedback signals can be designed and applied from either side of the electrodes or from both, which can induce different control performances. Those discussions can be found in Rahim and Younis (2016). Here, we only discuss one situation, as shown in Figure 1: (1) DC voltages applied on both electrodes is equal and AC voltage is only applied on the upper electrode; (2) delayed velocity feedback signals, treated as feedback voltage, are applied on the upper electrode, as mentioned by Shao et al. (2013). In addition, electrostatic force without a fringing effect is consdiered here, which has been illustrated by Younis (2011) and adopted by Haghighi and Markazi (2010) and Han et al. (2015a, 2015b). Thus, this simplification seems to be valid to qualitatively investigate the effect of delayed velocity feedback control on the stability and perturbed dynamics of a MEMS resonator.

Under the above considerations, the electrostatic force F_e with delayed velocity feedback control can be expressed as (Mestrom et al., 2008; Shao et al., 2013)

F_{e} = \frac{1}{2} \frac{C_{0} d_{0}}{(d_{0} - \hat{x})^{2}} [V_{dc} + V_{ac} \sin (Ω t) + G (\hat{x}' - \hat{x}')]^{2} - \frac{1}{2} \frac{C_{0} d_{0}}{(d_{0} + \hat{x})^{2}} V_{dc}^{2}

(1)

where (′) represents the derivative with respect to time t,

C_{0} = ɛ_{d} A_{s} / d_{0}

is the capacitance over the gap when

\hat{x} = 0

ɛ_{d}

is the dielectric constant of the air, A_s is the electrode area,

\hat{x}'_{d} = \hat{x}' (t - t_{d})

, t_d is the delay time and G is the delayed velocity feedback gain.

Combined with the work of Shao et al. (2013) and Han et al. (2015a) and taking equation (1) into consideration, the governing equation of motion of this dynamic system can be expressed as

m \hat{x} ″ + c \hat{x}' + k_{1} \hat{x} + k_{3} {\hat{x}}^{3} = F_{e}

(2)

where m, c, k₁ and k₃ are the effective mass, damping, linear and cubic stiffness, respectively.

It should be noted that although equation (2) seems to be similar to that of Shao et al. (2013), it actually has two obvious differences. One is the consideration of microbeam mid-plane stretching (nonlinear cubic stiffness), which is more suitable for describing the structural property of doubly clamped microbeams. The other is the expression of electrostatic force generated by two symmetrical electrodes. Thus, our dynamic model is different from that of Shao et al. (2013) to some extent.

For convenience, we introduce the following nondimensional variables

τ = ω_{0} t, τ_{d} = ω_{0} t_{d}, x = \frac{\hat{x}}{d_{0}}, x_{d} = \frac{{\hat{x}}_{d}}{d_{0}}

(3)

where

ω_{0} = \sqrt{k_{1} / m}

is the natural angular frequency of the microbeam.

Substituting equation (3) into equation (2) yields

\overset{··}{x} + μ \overset{\cdot}{x} + x + α x^{3} = γ \frac{[1 + ρ \sin (ω τ) + G_{V} ({\overset{\cdot}{x}}_{d} - \overset{\cdot}{x})]^{2}}{(1 - x)^{2}} - γ \frac{1}{(1 + x)^{2}}

(4)

where (ċ) represents the derivative with respect to nondimensional time τ and

μ = \frac{c}{\sqrt{k_{1} m}}, α = \frac{k_{3} d_{0}^{2}}{k_{1}}, γ = \frac{C_{0} V_{dc}^{2}}{2 k_{1} d_{0}^{2}}, ρ = \frac{V_{ac}}{V_{dc}}, ω = \frac{Ω}{ω_{0}}, G_{V} = \frac{G ω_{0} d_{0}}{V_{dc}}

The equilibrium position

x_{e} (| x_{e} | < 1)

of the Hamiltonian system of equation (4) due to DC voltage is summarized in Table 1 (Han et al., 2015a, 2015b), in which

ℜ_{1}

and

ℜ_{2}

can be expressed as

where

q_{0} = \frac{2 [(1 + α)^{3} - 54 α^{2} γ]}{27 α^{3}}

Δ_{0} = \frac{4 γ [27 α^{2} γ - (1 + α)^{3}]}{27 α^{4}}

ϖ = \frac{- 1 + \sqrt{3} i}{2}

i = \sqrt{- 1}

Table 1.

Expression and stability of equilibrium positions under different α and γ (S: stable; US: unstable).

Case	Equilibrium position and stability
$0 < α < 2$ and $0 < γ < 0.25$	$x_{e, 0} = 0 (S)$ , $x_{e, 2}^{\pm} = \pm ℜ_{2} (US)$
$0 < α < 2$ and $γ > 0.25$	$x_{e, 0} = 0 (US)$
$α > 2$ and $0 < γ < 0.25$	$x_{e, 0} = 0 (S)$ , $x_{e, 2}^{\pm} = \pm ℜ_{2} (US)$
$α > 2$ and $0.25 < γ < (1 + α)^{3} / (27 α^{2})$	$x_{e, 0} = 0 (US)$ , $x_{e, 1}^{\pm} = \pm ℜ_{1} (S)$ , $x_{e, 2}^{\pm} = \pm ℜ_{2} (US)$
$α > 2$ and $γ > (1 + α)^{3} / (27 α^{2})$	$x_{e, 0} = 0 (US)$

In addition, potential energy and phase portraits of the Hamiltonian system are qualitatively depicted in Figure 2. The dynamic properties can be easily grasped from these figures. Obviously, only the situations in Figures 2(a), (c) and (d) have stable centers. Under the situations in Figures 2(b) and (e), no stable equilibrium position exists. The microresonator undergoes the pull-in instable phenomenon. All these figures coincide with the theoretical results in Table 1.

Figure 2.

Potential energy and phase portraits of the corresponding Hamiltonian system. (a) α = 1, γ = 0.1, (b) α = 1, γ = 0.3, (c) α = 3, γ = 0.1, (d) α = 3, γ = 0.255, (e) α = 3, γ = 0.27.

3. Theoretical analysis

In this section, vibration properties of this microresonator with delayed velocity feedback control are theoretically investigated.

3.1. Time delay stability

Generally, harmonic AC voltage applied on MEMS resonators is relatively smaller than DC voltage. Hence, AC excitation can be treated as small disturbances to the MEMS resonator and is approximately ignored here for time delay stability analysis. We introduce $x = x_{e} + u$ , where u is the dynamic amplitude of the motion. Assume the feedback gain is smaller than the DC voltage (Shao et al., 2013), and then the unexcited system of equation (4) can be linearized as

\overset{··}{u} + (μ + g) \overset{\cdot}{u} - g {\overset{\cdot}{u}}_{d} + ω_{n}^{2} u = 0

(7)

where

ω_{n} = \sqrt{1 + 3 α x_{e}^{2} - \frac{2 γ}{(1 - x_{e})^{3}} - \frac{2 γ}{(1 + x_{e})^{3}}}

is the equivalent natural angular frequency of system (4),

g = \frac{2 γ G_{V}}{(1 - x_{e})^{2}}

The characteristic equation of equation (7) can be given by

Θ (s, τ_{d}) \equiv s^{2} + (μ + g - {ge}^{- s τ_{d}}) s + ω_{n}^{2} = 0

(8)

According to Hu and Wang (2002), linearized system (7) is delay-independent stable if all the roots of $Θ (s, 0) = 0$ have negative real parts and the critical condition $Θ (i η, τ_{d}) = 0$ has no real root η for any given $τ_{d}$ . Based on the Routh–Hurwitz criterion, the first condition yields $μ > 0$ and $ω_{n}^{2} > 0$ . Obviously, the system parameters satisfy this condition. The second condition is somewhat complicated, which needs to be investigated further. Assume that $s = i η$ $(η > 0)$ is an imaginary root of equation (8). Substituting s into equation (8) and separating the real and imaginary parts yields

{\sin (η τ_{d}) = \frac{ω_{n}^{2} - η^{2}}{g η} \cos (η τ_{d}) = \frac{μ + g}{g}

(9)

Eliminating $τ_{d}$ from equation (9), one can obtain

η^{4} + p η^{2} + q = 0

(10)

where

p = μ^{2} + 2 μ g - 2 ω_{n}^{2}

q = ω_{n}^{4}

Based on the above results and Hu and Wang (2002), linearized system (7) is delay-independent stable if p and q satisfy the following conditions

p \geq 0, q \geq 0 or p < 0, p^{2} - 4 q < 0

(11)

After some calculation, it can be easily derived that if μ and g meet the following condition

2 g + μ > 0

(12)

then the stability of linearized system (7) is delay-independent, which means the system is stable for all

τ_{d} > 0

Next, the situation with $2 g + μ < 0$ is discussed. Two positive real roots of equation (10) can be derived as

η_{1, 2} = \sqrt{\frac{- p \pm \sqrt{p^{2} - 4 q}}{2}}

(13)

It is easy to prove that $η_{1}^{2} > ω_{n}^{2}$ and $η_{2}^{2} < ω_{n}^{2}$ . Thus, a series of critical time delays (Wang and Hu, 2003) can be derived from equation (9), which are given by

τ_{d, 1}^{k} = \frac{1}{η_{1}} [\arccos (\frac{μ + g}{g}) + 2 k π], k = 0, 1, 2 \dots

(14)

τ_{d, 2}^{k} = \frac{1}{η_{2}} [2 π - \arccos (\frac{μ + g}{g}) + 2 k π], k = 0, 1, 2 \dots

(15)

Next, let $s (τ_{d}) = ξ (τ_{d}) + i η (τ_{d})$ be a root of equation (8). One can obtain from equations (8) and (9) that

Re {[s'_{τ_{d}} (τ_{d, 1}^{k})]^{- 1}} = + \frac{\sqrt{p^{2} - 4 q}}{g^{2} η_{1}^{2}} > 0, Re {[s'_{τ_{d}} (τ_{d, 2}^{k})]^{- 1}} = - \frac{\sqrt{p^{2} - 4 q}}{g^{2} η_{2}^{2}} < 0

(16)

where

s'_{τ_{d}} (τ_{d})

is the derivative of

s (τ_{d})

with respect to delay time

τ_{d}

and

Re (\cdot)

is the real part of variable.

Note that $ξ'_{τ_{d}} (τ_{d, i}^{k})$ has the same sign as $Re {[s'_{τ_{d}} (τ_{d, i}^{k})]^{- 1}}$ . Thus, based on Hopf bifurcation theory, $τ_{d, 1}^{k}$ and $τ_{d, 2}^{k}$ are series of Hopf critical points. The stability of zero equilibrium position of linearized system (8) can be decided according to Wang and Hu (2000). When $τ_{d} \in [0, τ_{d, 1}^{0})$ , all the roots of equation (8) are in the left-hand complex plane and the zero equilibrium point is stable. If $τ_{d, 1}^{1} > τ_{d, 2}^{0}$ and $τ_{d} \in (τ_{d, 1}^{0}, τ_{d, 2}^{0})$ , at least one root of equation (8) is in the right-hand complex plane and the zero equilibrium position is unstable, and so on. That is to say, when delay time $τ_{d}$ passes though the threshold $τ_{d, 1}^{k}$ from left to right, supercritical Hopf bifurcation happens. If $τ_{d}$ passes though the threshold $τ_{d, 2}^{k}$ from left to right, then subcritical Hopf bifurcation happens. It should be noted that the stability switches (Wang and Hu, 2000) may happen with the increase of k. For more details, see Wang and Hu (2000).

Stability charts depicting nondimensional time delay $τ_{d}$ versus feedback gain g with different system parameters are shown qualitatively in Figure 3, in which $T_{n} = 2 π / ω_{n}$ . Given that g is variable, there must be an intersection $((k + 1) T_{n}, g_{c, k})$ between $τ_{d, 2}^{k}$ and $τ_{d, 1}^{k + 1}$ . $g < g_{c, k}$ leads to $τ_{d, 1}^{k + 1} < τ_{d, 2}^{k}$ , then stability switches occur. Thus, only the region above $τ_{d, 1}^{k}$ and $τ_{d, 2}^{k}$ is stable, otherwise supercritical or subcritical Hopf bifurcation happens. For instance, numerical simulations of equation (4) about Point A and B in Figure 3(a) are shown in Figure 4. From this figure, one can see that when control parameters are in the stable region, the MEMS resonator is dynamically stable. If the control parameters are in the unstable region (point B), the system results in dynamic pull-in instability. Here, Hopf bifurcation is assumed to be undesirable, as it may greatly change the vibration property of the original dynamic system (dynamic pull-in). Therefore, Hopf bifurcation regions are regarded as unstable regions.

Figure 3.

Stability chart with μ = 0.02 (Τ_n = 2π/ω_n): (a) α = 0.3, γ = 0.01, the origin is a stable equilibrium position; (b) α = 3, γ = 0.26, the origin is an unstable equilibrium position.

Figure 4.

Displacement response of the resonator with ρ = 0.01, ω = 1, τ_d = 6.95T_n: (a) g = −0.03; (b) g = −0.08.

3.2. Perturbation solution

In this section, the small vibration of equation (4) around different equilibrium positions is investigated. Assume the feedback control satisfies the stable condition (Figure 3). Consider the following terms (Shao et al., 2013): $u = O (ɛ)$ , $ρ = O (ɛ^{3})$ , $g = O (ɛ^{2})$ , $μ = O (ɛ^{2})$ , where ɛ is regarded as a small nondimensional bookkeeping parameter (it is just a symbol for distinguishing small and nonsmall parameters, without any true value). Scaling the damping and expanding the electro-dynamic term of equation (4) up to the third order of ɛ yields

\overset{··}{u} + ɛ^{2} μ_{0} \overset{\cdot}{u} + ω_{n}^{2} u + a_{q} u^{2} + a_{c} u^{3} = ɛ^{3} f \sin (ω τ) + ɛ^{2} g {\overset{\cdot}{u}}_{d}

(17)

where

μ_{0} = μ + g

a_{q} = 3 α x_{e} - \frac{3 γ}{(1 - x_{e})^{4}} + \frac{3 γ}{(1 + x_{e})^{4}}

a_{c} = α - \frac{4 γ}{(1 - x_{e})^{5}} - \frac{4 γ}{(1 + x_{e})^{5}}

f = \frac{2 γ ρ}{(1 - x_{e})^{2}}

Here, the primary resonance is investigated (Nayfeh et al., 2007). To describe the nearness of ω to $ω_{n}$ , a detuning parameter σ is introduced and is defined by

ω = ω_{n} + ɛ^{2} σ

(18)

The approximate solution of equation (17) can be written as

u (τ; ɛ) = ɛ u_{1} (T_{0}, T_{1}, T_{2}) + ɛ^{2} u_{2} (T_{0}, T_{1}, T_{2}) + ɛ^{3} u_{3} (T_{0}, T_{1}, T_{2}) + \dots

(19)

where

T_{j} = ɛ^{j} τ

(j = 0, 1, 2)

By using the following differential operators based on Taylor expansion, the delay displacement term expanded up to third order of ɛ can be approximately written as (Nayfeh, 2011)

u_{d} (τ - τ_{d}; ɛ) \approx ɛ u_{1} (T_{0} - τ_{d}, T_{1}, T_{2}) + ɛ^{2} u_{2} (T_{0} - τ_{d}, T_{1}, T_{2}) - ɛ^{2} τ_{d} D_{1} [u_{1} (T_{0} - τ_{d}, T_{1}, T_{2})] + ɛ^{3} u_{3} (T_{0} - τ_{d}, T_{1}, T_{2}) - ɛ^{3} τ_{d} D_{2} [u_{1} (T_{0} - τ_{d}, T_{1}, T_{2})] - ɛ^{3} τ_{d} D_{1} [u_{2} (T_{0} - τ_{d}, T_{1}, T_{2})] + ɛ^{3} \frac{1}{2} τ_{d}^{2} D_{1}^{2} [u_{1} (T_{0} - τ_{d}, T_{1}, T_{2})] + \dots

(21)

then the delay velocity term can be calculated as

{\overset{\cdot}{u}}_{d} (τ - τ_{d}; ɛ) \approx (D_{0} + ɛ D_{1} + ɛ^{2} D_{2}) u_{d} (τ - τ_{d}; ɛ)

Applying the MMS (Nayfeh, 2011), one can obtain the average equation in polar form

{\frac{d a}{d τ} = - \frac{μ_{0}}{2} a - \frac{f}{2 ω_{n}} \cos ϕ + \frac{g}{2} a \cos (ω_{n} τ_{d}) a \frac{d ϕ}{d τ} = a σ + (\frac{5 a_{q}^{2}}{12 ω_{n}^{3}} - \frac{3 a_{c}}{8 ω_{n}}) a^{3} + \frac{f}{2 ω_{n}} \sin ϕ + \frac{g}{2} a \sin (ω_{n} τ_{d})

(22)

where

ϕ = σ \cdot τ - β

, a and β are the approximate amplitude and phase of u.

A steady-state response can be obtained by imposing the condition $\frac{d a}{d τ} = \frac{d ϕ}{d τ} = 0$ . Finally, the frequency response equation can be expressed as

a^{2} [\frac{μ_{e}^{2}}{4} + (ω - ω_{n, e} + κ a^{2})^{2}] = (\frac{f}{2 ω_{n}})^{2}

(23)

where

μ_{e} = μ + g [1 - \cos (ω_{n} τ_{d})]

ω_{n, e} = ω_{n} - \frac{g}{2} \sin (ω_{n} τ_{d})

κ = \frac{5 a_{q}^{2}}{12 ω_{n}^{3}} - \frac{3 a_{c}}{8 ω_{n}}

The primary resonance peak value of the vibration amplitude and the backbone curve can be decided by

a_{max} = \frac{f}{ω_{n} μ_{e}}

(24)

ω = ω_{n, e} - κ a_{max}^{2}

(25)

From equations (23)–(25), one can see the following: (1) delayed velocity feedback control may tune the effective damping coefficient $μ_{e}$ , while it may also tune the effective natural angular frequency $ω_{n, e}$ ; (2) delay time $τ_{d}$ can periodically change the values of $μ_{e}$ and $ω_{n, e}$ , and the period is equal to T_n; (3) proper decrease of $μ_{e}$ can increase the peak value $a_{max}$ ; (4) negative κ leads to a hardening-type phenomenon and a positive one leads to a softening-type phenomenon; if $κ = 0$ , linear-like behavior emerges (Han et al., 2015a); (5) the feedback control strategy does not qualitatively affect the hardening/softening/linear-like behavior, as mentioned in point (4).

The stability of steady-state motion $(a_{0}, ϕ_{0})$ can be determined through the linearized equations of equation (22) by letting $a = a_{0} + Δ a$ and $ϕ = ϕ_{0} + Δ ϕ$ , where $(Δ a, Δ ϕ)$ represent a small disturbance. Then one can obtain

{\frac{d Δ a}{d τ} \frac{d Δ ϕ}{d τ}} = [- \frac{μ_{e}}{2} \frac{f}{2 ω_{n}} \sin ϕ_{0} 2 κ a_{0} - \frac{f \sin ϕ_{0}}{2 ω_{n} a_{0}^{2}} \frac{f \cos ϕ_{0}}{2 ω_{n} a_{0}}] {Δ a Δ ϕ}

(26)

Thus, the stability of steady-state motion depends on the eigenvalues of the coefficient matrix on the right-hand sides of equation (26). The eigenvalue equation can be decided by

| λ + \frac{μ_{e}}{2} - \frac{f}{2 ω_{n}} \sin ϕ_{0} - 2 κ a_{0} + \frac{f \sin ϕ_{0}}{2 ω_{n} a_{0}^{2}} λ - \frac{f \cos ϕ_{0}}{2 ω_{n} a_{0}} | = 0

(27)

which can be simplified as

λ^{2} + δ_{1} λ + δ_{2} = 0

(28)

where

δ_{1} = μ_{e}

δ_{2} = μ_{e}^{2} / 4 + [ω - ω_{n, e} + κ a_{0}^{2}] \times [ω - ω_{n, e} + 3 κ a_{0}^{2}]

Hence, the steady-state motions are stable when both the following conditions are satisfied

{δ_{1} > 0 δ_{2} > 0

(29)

From the expressions of $δ_{1}$ and $δ_{2}$ , one can see that condition $δ_{1} > 0$ , that is, $μ_{e} > 0$ , guarantees the existence of perturbation solution a₀, while $δ_{2} > 0$ is used to predict the stability of multiple solutions a₀. Only if all the conditions are satisfied is the perturbation solution stable.

3.3. Stability analysis

In light of the analysis in Sections 3.1 and 3.2, one can see that the existence of small vibration u depends on two conditions: (1) time delay stability (Figure 3) and (2) $μ_{e} > 0$ . One knows that the increase of $τ_{d}$ can only periodically change $μ_{e}$ and $ω_{n, e}$ . Thus, time delay analysis in region $g \in (g_{c, k}, + \infty)$ and $τ_{d} \in [{kT}_{n}, (k + 1) T_{n}]$ $(k = 0, 1, 2 \dots)$ is enough to grasp the dynamic characteristic of the system. However, periodic increase of $τ_{d}$ can decrease the stable region (Figure 3). When $k = 0$ , the stable region is the largest. Therefore, in the following analysis, $k = 0$ is adopted.

From the expression of g, one can see that for a certain g, DC voltage $V_{dc}$ can only affect the relative size of the feedback gain. Under different parameters, the stability charts should be in qualitative agreement with each other. Thus, a case study is carried out to attempt to investigate the effect of time delay on the qualitative stability of the resonator. Based on the stability analysis in Sections 3.1 and 3.2, a final stability chart is shown in Figure 5. In this figure, five conditions, $0 < μ_{e} < μ$ , $τ_{d, 1}^{0}$ , $τ_{d, 2}^{0}$ , $τ_{d} = T_{n} / 2$ and $μ + 2 g = 0$ , divide the time delay space $τ_{d} - g$ into nine regions. In each stable region, the effective damping and natural angular frequency follow some certain conditions. For example, the control parameter in region R-1 indicates that the effective damping is greater than that of the uncontrolled system, while the effective natural angular frequency is less than that of the uncontrolled system. Therefore, feedback control in R-1 can finally increase system damping while decreasing system natural frequency. Typically, Lines L-1 and L-2 correspond to two critical states with $τ_{d} = T_{n} / 2$ , which have been studied by Shao et al. (2013) based on a one-electrode microresonator. Under this situation, feedback control can only increase (L-1) or decrease (L-2) the effective damping. Here, it is noted that our work focuses on the entire stable regions and the corresponding control properties, all of which can be found seen Figure 5. As case studies, in our following study, $τ_{d} = T_{n} / 4$ and $τ_{d} = 3 T_{n} / 4$ are selected to verify the existence of frequency and damping trimming phenomena of the system.

Figure 5.

Stability chart and numerical validation for small vibration in $τ_{d} \in [0, T_{n}]$ with α = 0.3, μ = 0.02, γ = 0.01: (a) stability chart; (b) numerical validation with equation (4). S: stable; US: unstable.

Another important note is that the existence condition of the periodic solution via perturbation analysis ( $δ_{1} > 0$ , i.e. $μ_{e} > 0$ ) overestimates the system stable region. Two important unstable regions, R-7 and R-8, are ignored by this existence condition, but are considered by the time delay stability estimation. In fact, besides R-9, R-7 and R-8 are unstable as well. The numerical results of equation (4) when control parameters are in these regions are shown in Figure 5(b). From these figures, one can see that all three regions are unstable. Their only difference is that the dynamic pull-in time is longer in R-9 than that in R-7 or R-8. The delayed stability condition via linearized analysis seems to be more suitable for stability estimation. As long as the delay stability condition in Section 3.1 is satisfied, the dynamic system is stable under small amplitude vibration.

4. Results

In this section, a specific doubly clamped microbeam (Table 2) with different initial gap widths (Bumkyoo and Lovell, 1997) is considered to investigate the effect of delayed velocity feedback control on the vibration properties of the system. The primary resonance peak value $a_{max}$ is assumed to be much smaller than h in light of the small amplitude assumption in Section 3. Under the first mode assumption with classical beam theory, equivalent parameters of the resonator can be obtained as (Kaajakari et al, 2004) $m = 0.396 ρ_{m} whl$ , $k_{1} = 125.1 EI / l^{3}$ , $k_{3} = 0.767 k_{1} / h^{2}$ , where $I = {wh}^{3} / 12$ is the moment of inertia. Then, α is equal to (1) Case-1: 0.767; (2) Case-2: 3.068. Based on Table 1 and our previous study (Han et al., 2015a), one can see the following.

In Case-1, the system has only one stable equilibrium position $x_{e, 0} = 0$ when the tuning range of DC voltage is $V_{dc} \in (0 V, 15.587 V)$ . At this point, $κ = \frac{3 (8 γ - α)}{8 \sqrt{1 - 4 γ}}$ . $κ = 0$ leads to a critical DC voltage of 9.653 V. If the actual DC voltage satisfies $0 V < V_{dc} < 9.653 V$ , the system exhibits hardening-type behavior; else if $V_{dc} = 9.653 V$ , linear-like behavior, else, softening-type behavior.

In Case-2, the system has (a) only one stable equilibrium position, $x_{e, 0} = 0$ , when $0 V < V_{dc} < 44.087 V$ ; (b) two stable equilibrium positions, $x_{e, 1}^{\pm} = \pm ℜ_{1}$ , when $44.087 V < V_{dc} < 45.381 V$ . Further analysis shows that parameter κ is always negative in situation (a), yielding hardening-type behavior, and positive in situation (b), yielding softening-type behavior.

Table 2.

Parameters of a doubly clamped microbeam (Bumkyoo and Lovell, 1997).

Parameters	Values
Length l, width w and thickness h (µm)	400 × 45 × 2
Density $ρ_{m}$ (kg/m³)	2.33 × 10³
Dielectric constant of the air $ɛ_{d}$ (F/m)	8.85 × 10⁻¹²
Young’s modulus E (N/m²)	1.65 × 10¹¹
Damping coefficient c (kg/s)	8.96 × 10⁻⁸
Natural frequency f₀ ( $2 π f_{0} = ω_{0}$ ) (kHz)	85.898
Initial gap width d₀ (µm)	Case-1: 2; Case-2: 4

4.1. Frequency response

Firstly, we focus on the vibration characteristics in the neighborhood of zero equilibrium position $x_{e, 0} = 0$ . The analytical results (equation (23) in dimensional form) obtained by the MMS and the numerical results obtained by the long time integration (LTI) of equation (2) are shown in Figures 6 –9. Notice that the excitation frequency f_e and angular frequency Ω satisfy $Ω = 2 π f_{e}$ and $\hat{u} = u \cdot d_{0}$ . It is obvious from these figures that in Case-1, the vibrations of the resonator gradually exhibit hardening-type, linear-like or softening-type behaviors with the increasing of DC voltage (Figures 6 –8). In Case-2, only hardening-type behavior exists (Figure 9). When delay time $τ_{d} = T_{n} / 2$ , $μ_{e} = μ + 2 g$ and $ω_{n, e} = ω_{n}$ . Then the variation of feedback gain G can only change the effective damping $μ_{e}$ . Positive G can increase $μ_{e}$ , while negative G can decrease $μ_{e}$ (Figures 6(a)–9(a)). These results coincide with the former analysis (L-1 and L-2 in Figure 5). In Shao et al. (2013), this characteristic was found in a one-electrode microresonator system. When delay time $τ_{d} = T_{n} / 4$ , $μ_{e} = μ + g$ and $ω_{n, e} = ω_{n} - g / 2$ . Positive G can increase $μ_{e}$ , while it can decrease $ω_{n, e}$ (R-1 in Figure 5). Negative G can decrease $μ_{e}$ while it can increase $ω_{n, e}$ (R-3 and R-5 in Figure 5). From Figures 6(b)–9(b), one can see that the analytical analysis in Section 3.3 is correct. On the contrary, when $τ_{d} = 3 T_{n} / 4$ , $μ_{e} = μ + g$ and $ω_{n, e} = ω_{n} + g / 2$ . Positive G can increase $μ_{e}$ and $ω_{n, e}$ (R-2 in Figure 5). Negative G can decrease $μ_{e}$ and $ω_{n, e}$ (R-4 and R-6 in Figure 5). These can be confirmed from Figures 6(c)–9(c).

Figure 6.

Hardening-type behavior in Case-1 around x_e_,0 with V_dc = 7 V, V_ac = 40 mV; the unit of G is mV/s. (a) τ_d = T_n/2. (b) τ_d = T_n/4. (c) τ_d=3T_n/4. MMS: method of multiple scales; LTI: long time integration.

Figure 7.

Linear-like behavior in Case-1 around x_e_,0 with V_dc = 9.653 V, V_ac = 20 mV; the unit of G is mV/s. (a) τ_d = T_n/2. (b) τ_d = T_n/4. (c) τ_d=3T_n/4. MMS: method of multiple scales; LTI: long time integration.

Figure 8.

Softening-type behavior in Case-1 around x_e_,0 with V_dc = 12 V, V_ac = 10 m V; the unit of G is mV/s. (a) τ_d = T_n/2. (b) τ_d = T_n/4. (c) τ_d=3T_n/4. MMS: method of multiple scales; LTI: long time integration.

Figure 9.

Hardening-type behavior in Case-2 around x_e_,0 with V_dc = 35 V, V_ac = 20 mV; the unit of G is mV/s. (a) τ_d = T_n/2. (b) τ_d = T_n/4. (c) τ_d = 3T_n/4. MMS: method of multiple scales; LTI: long time integration.

Next, the vibration characteristics around two equilibrium positions $x_{e, 1}^{\pm} = \pm ℜ_{1} = \pm 0.227$ ( $V_{dc} = 45 V$ ) are investigated. According to our previous works (Han et al., 2015a), one can see from equations (7) and (17) that $ω_{n} |_{x_{e} = x_{e, 1}^{+}} = ω_{n} |_{x_{e} = x_{e, 1}^{-}}$ , $f |_{x_{e} = x_{e, 1}^{+}} > f |_{x_{e} = x_{e, 1}^{-}}$ and $g |_{x_{e} = x_{e, 1}^{+}} > g |_{x_{e} = x_{e, 1}^{-}}$ . Under the same AC excitation and feedback control the following applies: (1) the equivalent natural angular frequency should be invariable around $x_{e, 1}^{+} = + ℜ_{1}$ or $x_{e, 1}^{-} = - ℜ_{1}$ ; (2) the amplitude of the resonator should be larger around $x_{e, 1}^{+} = + ℜ_{1}$ than that around $x_{e, 1}^{-} = - ℜ_{1}$ ; (3) the frequency trimming value is relatively larger around $x_{e, 1}^{+} = + ℜ_{1}$ than that around $x_{e, 1}^{-} = - ℜ_{1}$ . As is shown in Figures 10 and 11, our analytical results (MMS) show excellent agreement with the numerical results (LTI), which verify all these points.

Figure 10.

Softening-type behavior in Case-2 around $x_{e, 1}^{+}$ with V_dc = 45 V, V_ac = 4 mV; the unit of G is mV/s. (a) τ_d = T_n/2 (b) τ_d = T_n/4. (c) τ_d = 3T_n/4. MMS: method of multiple scales; LTI: long time integration.

Figure 11.

Softening-type behavior in Case-2 around $x_{e, 1}^{-}$ with V_dc = 45 V, V_ac = 4 mV; the unit of G is mV/s. (a) τ_d = T_n/2 (b) τ_d = T_n/4. (c) τ_d = 3T_n/4. MMS: method of multiple scales; LTI: long time integration.

In this section, we study the nonlinear characteristics of the resonator and reveal the dynamic properties with delayed feedback control through case studies (Table 2). In addition, frequency and damping trimming properties are investigated as well to verify our analytical results in Section 3. These phenomena seem to be attractive to MEMS designers. In our following study, we will discuss these aspects in detail and offer some possible design tips.

4.2. Frequency and damping trimming

In fact, damping trimming can occur without frequency trimming when $τ_{d} = T_{n} / 2$ . Otherwise, it is accompanied with frequency trimming. In this section, both of these phenomena are investigated.

Firstly, we must emphasize three different natural frequencies: (1) natural frequency of the resonator without DC voltage f₀ (the unit is Hz), where $f_{0} = ω_{0} / (2 π)$ ; (2) equivalent natural frequency of the resonator with DC voltage f_n (the unit is Hz), where $f_{n} = ω_{0} ω_{n} / (2 π)$ ; (3) effective natural frequency of the resonator with DC voltage and feedback control $f_{n, e}$ (the unit is Hz), where $f_{n, e} = ω_{0} ω_{n, e} / (2 π)$ . The first is related to the design dimension of the MEMS resonator, which determines the original natural frequency of the resonator. The second mainly depends on the deflected DC voltage, which can directly tune the natural frequency of the resonator. The last can be seen as the result combined with DC voltage and feedback control.

According to the above relationships, one can obtain the variation of equivalent natural frequency versus DC voltage under different initial gap widths, which is shown in Figure 12. In Case-1, the equivalent natural frequency f_n decreases with the increasing of DC voltage. Pull-in instability is triggered when $f_{n} = 0$ . In Case-2, f_n decreases firstly with the increase of $V_{dc}$ until it is equal to zero. On continuing to increase $V_{dc}$ , f_n increases at first and then quickly decreases until pull-in happens. From the viewpoint of MEMS vibration, the resonator in Case-1 can only vibrate in the neighborhood of the zero equilibrium position. The natural frequency decreases with the increase of DC voltage. However, the resonator in Case-2 may also vibrate around the two new centers besides the zero equilibrium position. The natural frequency exhibits a rebounding phenomenon. Furthermore, the pull-in voltage in Case-1 is smaller than that in Case-2.

Figure 12.

The variation of equivalent natural frequency versus direct current (DC) voltage.

Assume the delayed feedback control does not change the vibration property. For convenience, we only investigate the delay-independent situation with $g > - μ / 2$ in order to adjust all delay times $τ_{d} \in (0, T_{n})$ . The simulation parameters are the same as those shown in Table 2. As the frequency and damping trimming properties in the neighborhood of equilibrium positions are similar to each other, we only take $V_{dc} = 6 V$ in Case-1 as a case study, which yields $G > - 187 mV / s$ . It is obvious from Figure 13 that the sign of feedback gain G has two contrary effects on frequency trimming. When $τ_{d} \in (0, T_{n} / 2)$ , positive G can shift the natural frequency to the left, while negative G can shift it to the right. When $τ_{d} \in (T_{n} / 2, T_{n})$ , positive G can shift the natural frequency to the right, while negative G can shift it to the left. Unlike frequency trimming, damping trimming is always positive when $G > 0$ and negative when $G < 0$ . Notice that when $τ_{d} = 0$ or $τ_{d} = T_{n}$ , both these phenomena disappear. When $τ_{d} = T_{n} / 2$ , only frequency trimming disappears. According to the above analysis, one can conclude that these two characteristics can be expanded to the cases with $τ_{d} \in (k, {kT}_{n} / 2)$ , $τ_{d} \in ({kT}_{n} / 2, {kT}_{n})$ or $τ_{d} = {kT}_{n} / 2$ , where $k = 1, 2, 3 \dots$ .

Figure 13.

Frequency and damping trimming in Case-1 around x_e_,0 with V_dc = 6 V; the unit of G is mV/s. (a) Frequency trimming and (b) Damping trimming.

4.3. Optimum operating DC voltage

The linear-like state (Figure 7) seems to be more attractive to MEMS designers (Han et al., 2015a) because one can use delayed velocity feedback control to realize system frequency/damping trimming and, hence, enable the operation of MEMS resonators at a larger amplitude and more powerful signal (Alsaleem and Younis, 2010, 2011). Under this vibration state, one must meet the following condition: $0 < α < 2$ , which is equivalent to $d_{0} / h < 1.615$ . To the best of our knowledge, most ordinary electrostatically actuated microbeams satisfy $d_{0} / h \leq 1$ . Thus, this section will mainly focus on this vibration state.

In Figure 7, one can see that the analytical results (MMS, equation (23) in dimensional form) agree well with the numerical results (LTI, equation (2)) when the amplitude of the resonator $\hat{u}$ is much smaller than the thickness of the microbeam. Theoretically, a linear-like vibration state exists if α and γ meet $α = 8 γ$ , which yields the expression of optimum operating DC given by

V_{dc} \approx \frac{1.414 d_{0}^{2}}{l^{2}} \sqrt{\frac{{Ehd}_{0}}{ɛ_{d}}}

(30)

and the equivalent natural frequency f_n (the unit of f_n is Hz) can be written as

f_{n} \approx \frac{0.817}{l^{2}} \sqrt{\frac{E (h^{2} - 0.384 d_{0}^{2})}{ρ_{m}}}

(31)

Based on equations (30) and (31), one can obtain the optimum operating DC voltage (solid lines) and its equivalent natural frequency (dashed lines) for different microbeam-based resonators. Figure 14 shows some optimum design curves depicting these two parameters versus the initial gap width. It is obvious from Figure 14(a) that the increase of the length of the microbeam can decrease the available design region of DC voltage and its equivalent natural frequency. However, the cut-off line (dot-dashed line) is invariable, as the critical initial gap width $d_{0} = 1.615 h = 3.23 μ m$ is a constant value. In Figure 14(b), the design region increases with the increase of thickness of the beam. Furthermore, the cut-off line moves to the right, which means an increase of the critical initial gap width. Perhaps these two figures can be used as a tool to approximately design a linear-like vibration state of a specific resonator. For example, assume that the working frequency of a predesigned resonator is about 60 kHz. If the dimension of the microbeam is invariable, one can derive from Figure 14(a) that the optimum parameters satisfy $d_{0} = 2.31 μ m$ and $V_{dc} = 13.851 V$ . Note that the relative values of $V_{ac}$ , G and $τ_{d}$ can be first evaluated by equation (24). Thus, delayed velocity feedback control can be used to realize the fine-tuning of frequency and damping. As a case study, the analytical predictions via equation (23) in dimensional form and numerical results via equation (2) are shown in Figure 15, which verify the correctness of our predictions. Here, it must be noted that the expressions of optimum DC voltage and the equivalent natural frequency are both approximate solutions. Therefore, they can only evaluate relatively accurate values, which can be viewed in Figure 15.

Figure 14.

Optimum operating direct current (DC) voltage (solid line) and its corresponding equivalent natural frequency (dashed line) versus the initial gap width d₀: (a) under different lengths l; (b) under different thicknesses h.

Figure 15.

Design case with feedback gain G = −50 mV/s (R: V_ac = 19 mV, τ_d = T_n/4; Z: V_ac = 25 mV, τ_d = 0; L: V_ac = 19 mV, τ_d = 3T_n/4). MMS: method of multiple scales; LTI: long time integration.

5. Discussion and conclusions

This paper established a simplified 1-DOF model to describe a doubly clamped microbeam actuated by two symmetrically located electrodes with delayed velocity feedback control. Through the analysis of the linearized time-delayed equation, the stable region depicting delay time versus feedback gain beyond Hopf bifurcation was obtained. Different from the periodic stability description as mentioned by Shao et al. (2013), we found that the stable region was nonperiodic and decreased with the increase of period k. Stability switches (Wang and Hu, 2000) do exist in this dynamic system. Then, under the small vibration assumption, an approximate analytical solution to characterize the nonlinear dynamics of the system was obtained by the application of the MMS. Two critical conditions to guarantee the existence and stability of small vibration solutions were deduced. Following this, a stability chart with k = 0, considering Hopf bifurcation and the above two critical conditions, was drawn. We found that the existence condition $δ_{1} > 0$ in equation (29) overestimated the size of the stable region (Figure 5), and demonstrated that R-7–R-9 needed to be avoided in practice. All of these analyses broaden our knowledge of the stability of this type of microresonator with delayed velocity feedback.

We also investigated the hardening, softening and linear-like behaviors of the resonator with feedback control through case studies. What we found was that the feedback control could not qualitatively affect the above dynamic properties. However, both the delay time and feedback gain could tune the effective damping and natural frequency of the system. Then, we discussed the frequency and damping trimming properties in detail and concluded some qualitative trimming tips. Furthermore, we focused our attention on the linear-like behavior and derived two formulas to estimate the optimum operating DC voltage and its corresponding natural frequency. Through a case study, the designed initial gap width and DC voltage were obtained according to the predesigned working frequency of a specific resonator. The above analysis demonstrates that DC voltage can be used to tune the equivalent natural frequency and delayed velocity feedback control can realize the fine-tuning of frequency and damping.

The entire investigation may contain some potential applications in dynamic MEMS devices (Alsaleem and Younis, 2010, 2011). For example, one can obtain the control parameter scopes against dynamic instability, and then increase system stability via damping trimming, changing the resonance frequency via frequency trimming. In addition, linear-like design and control analysis have potential in generating a more powerful signal, thus improving the MEMS resonator performance in mass sensing.

Here, it should be noted that all of the analytical results are obtained based on the small vibration assumption. Both the stability analysis via linearized assumption and the dynamic analysis via perturbation approach can be effective only for weakly nonlinear systems and small vibrations in the neighborhood of the equilibrium position. Undergoing large motion, other feasible analysis methods such as the Shooting Method or the Finite Difference Method should be chosen for dynamic and control analyses (Masri et al., 2015). Moreover, many other factors, such as the fringing effect, residue stress, squeeze film damping, and the scale effect, which may affect system dynamics, are not taken into account here during the analysis. These factors can indeed affect the performance of the system to some extent and need to be further investigated. Although our analysis of the 1-DOF model can only provide an approximate understanding of this system in a relatively simple way, it can actually provide some references for the subsequent analysis. Guided by these analytical results, further research can be carried out to focus on the distributed model and to investigate the effects of more complicated impact factors on system static or dynamic behaviors.

Footnotes

Declaration of Conflicting Interests

The author(s) declared the following potential conflicts of interest with respect to the research, authorship, and/or publication of this article: This manuscript is our original unpublished work and it is not being submitted to any other journal for reviews. No data have been fabricated or manipulated (including images) to support our conclusions. The authors declare that we have no conflict of interest and agree to submit this manuscript to Journal of Vibration and Control. Besides, the manuscript has no research results involving human participants and/or animals.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was supported by the National Natural Science Foundation of China (grant number 11372210, 51405343, 11302149), the Tianjin Research Program of Application Foundation and Advanced Technology (grant number 15JCQNJC05000) and the Scientific Research Foundation of Tianjin University of Technology and Education (Grant number KYQD1701, KYQD16009).

References

Alsaleem

Younis

(2010) Stabilization of electrostatic MEMS resonators using a delayed feedback controller. Smart Materials and Structures 19(3): 035016.

Alsaleem

Younis

(2011) Integrity analysis of electrically actuated resonators with delayed feedback controller. Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME 133(3): 031011.

Batra

Porfiri

Spinello

(2007) Review of modeling electrostatically actuated microelectromechanical systems. Smart Materials and Structures 16(6): R23–R31.

Bumkyoo

Lovell

(1997) Improved analysis of microbeams under mechanical and electrostatic loads. Journal of Micromechanics and Microengineering 7(1): 24–29.

Chen

Dai

(2013) Nonlinear behavior and characterization of a piezoelectric laminated microbeam system. Communications in Nonlinear Science and Numerical Simulation 18(5): 1304–1315.

El-Bassiouny

(2006) Vibration control of a cantilever beam with time delay state feedback. Z Zeitschrift Fur Naturforschung A 61(12): 629–640.

Fang

(2013) A new approach and model for accurate determination of the dynamic pull-in parameters of microbeams actuated by a step voltage. Journal of Micromechanics and Microengineering 23(10): 045010.

Lin

(2003) Chaos, chaos control and synchronization of electro-mechanical gyrostat system. Journal of Sound and Vibration 259(3): 585–603.

Haghighi

Markazi

AHD

(2010) Chaos prediction and control in MEMS resonators. Communications in Nonlinear Science and Numerical Simulation 15(10): 3091–3099.

10.

Han

Zhang

Wang

(2015a) Static bifurcation and primary resonance analysis of a MEMS resonator actuated by two symmetrical electrodes. Nonlinear Dynamics 80(3): 1585–1599.

11.

Han

Zhang

Wang

(2015b) Design considerations on large amplitude vibration of a doubly clamped microresonator with two symmetrically located electrodes. Communications in Nonlinear Science and Numerical Simulation 22(1–3): 492–510.

12.

Wang

(1998) Stability analysis of damped SDOF systems with two time delays in state feedback. Journal of Sound and Vibration 214(2): 213–225.

13.

Wang

(2002) Dynamics of Controlled Mechanical Systems with Delayed Feedback, Berlin: Springer.

14.

Dowell

Virgin

(1998) Resonances of a harmonically forced Duffing oscillator with time delay state feedback. Nonlinear Dynamics 15(4): 311–327.

15.

Chang

Huang

(2004) Some design considerations on the electrostatically actuated microstructures. Sensors and Actuators A-Physical 112(1): 155–161.

16.

Kaajakari

Mattila

Oja

et al. (2004) Nonlinear limits for single-crystal silicon microresonators. Journal of Microelectromechanical Systems 13(5): 715–724.

17.

Kouravand

(2011) Design and modeling of some sensing and actuating mechanisms for MEMS applications. Applied Mathematical Modelling 35(10): 5173–5181.

18.

Krylov

Ilic

Schreiber

et al. (2008) The pull-in behavior of electrostatically actuated bistable microstructures. Journal of Micromechanics and Microengineering 18(5): 055026.

19.

Lakrad

Belhaq

(2010) Suppression of pull-in instability in MEMS using a high-frequency actuation. Communications in Nonlinear Science and Numerical Simulation 15(11): 3640–3646.

20.

Luo

ACJ

Wang

(2002) Chaotic motion in a micro-electro–mechanical system with non-linearity from capacitors. Communications in Nonlinear Science and Numerical Simulation 7(1–2): 31–49.

21.

Masri

Shao

Younis

(2015) Delayed feedback controller for microelectromechanical systems resonators undergoing large motion. Journal of Vibration and Control 21(13): 2604–2615.

22.

Mestrom

RMC

Fey

RHB

Nijmeijer

(2009) Phase feedback for nonlinear MEM resonators in oscillator circuits. IEEE/ASME Transactions on Mechatronics 14(4): 423–433.

23.

Mestrom

RMC

Fey

RHB

van Beek

JTM

et al. (2008) Modelling the dynamics of a MEMS resonator: simulations and experiments. Sensors and Actuators A-Physical 142(1): 306–315.

24.

Naik

Singru

(2011) Resonance, stability and chaotic vibration of a quarter-car vehicle model with time-delay feedback. Communications in Nonlinear Science and Numerical Simulation 16(8): 3397–410.

25.

Nayfeh

(2011) The Method of Normal Form, New York: John Wiley & Sons.

26.

Nayfeh

(2012) Time-delay feedback control of lathe cutting tools. Journal of Vibration and Control 18(8): 1106–1115.

27.

Nayfeh

Ouakad

Najar

et al. (2010) Nonlinear dynamics of a resonant gas sensor. Nonlinear Dynamics 59(4): 607–618.

28.

Nayfeh

Younis

Abdel-Rahman

(2007) Dynamic pull-in phenomenon in MEMS resonators. Nonlinear Dynamics 48(1–2): 153–163.

29.

Park

Chen

Lai

(2008) Energy enhancement and chaos control in microelectromechanical systems. Physical Review E-Statistical, Nonlinear, and Soft Matter Physics 77(2): 026210.

30.

Rahim

Younis

(2016) Control of bouncing in MEMS switches using double electrodes. Mathematical Problems in Engineering 2016: 3479752 (1–10).

31.

Rezaie

Jahed Motlagh

M-R

(2011) An adaptive delayed feedback control method for stabilizing chaotic time-delayed systems. Nonlinear Dynamics 64(1–2): 167–176.

32.

Seleim

Towfighian

Delande

et al. (2012) Dynamics of a close-loop controlled MEMS resonator. Nonlinear Dynamics 69(12): 615–633.

33.

Shao

Masri

Younis

(2013) The effect of time-delayed feedback controller on an electrically actuated resonator. Nonlinear Dynamics 74(1–2): 257–270.

34.

Siewe

Tchawoua

Rajasekar

(2012) Parametric resonance in the Rayleigh–Duffing oscillator with time-delayed feedback. Communications in Nonlinear Science and Numerical Simulation 17(11): 4485–4493.

35.

Song

Sun

(2013) Nonlinear and chaos control of a micro-electro-mechanical system by using second-order fast terminal sliding mode control. Communications in Nonlinear Science and Numerical Simulation 18(9): 2540–2548.

36.

Sun

Yang

et al. (2006) Inducing or suppressing chaos in a double-well Duffing oscillator by time delay feedback. Chaos, Solitons and Fractals 27(3): 705–714.

37.

Sun

Yang

et al. (2007) Effects of time delays on bifurcation and chaos in a non-autonomous system with multiple time delays. Chaos, Solitons and Fractals 31(1): 39–53.

38.

Towfighian

Heppler

Abdel-Rahman

(2011) Analysis of a chaotic electrostatic micro-oscillator. Journal of Computational and Nonlinear Dynamics 6(1): 011001.

39.

Wang

(2003) Remarks on the perturbation methods in solving the second-order delay differential equations. Nonlinear Dynamics 33(4): 379–398.

40.

Wang

(2004a) Global dynamics of a Duffing oscillator with delayed displacement feedback. International Journal of Bifurcation and Chaos 14(8): 2753–2775.

41.

Wang

(2004b) Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback. Acta Mechanica Sinica 20(4): 426–434.

42.

Wang

(2000) Stability switches of time-delayed dynamic systems with unknown parameters. Journal of Sound and Vibration 233(2): 215–233.

43.

Chung

Chan

(2007) An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks. SIAM Journal on Applied Dynamical Systems 6(1): 29–60.

44.

Yamasue

Hikihara

(2006) Control of microcantilevers in dynamic force microscopy using time delayed feedback. Review of Scientific Instruments 77(5): 53703.

45.

Yamasue

Kobayashi

Yamada

et al. (2009) Controlling chaos in dynamic-mode atomic force microscope. Physics Letters A 373(35): 3140–3144.

46.

Younesian

Sadri

Esmailzadeh

(2014) Primary and secondary resonance analyses of clamped–clamped micro-beams. Nonlinear Dynamics 76(4): 1867–1884.

47.

Younis

(2011) MEMS Linear and Nonlinear Statics and Dynamics, New York: Springer.

48.

Zaitsev

Shtempluck

Buks

et al. (2012) Nonlinear damping in a micromechanical oscillator. Nonlinear Dynamics 67(1): 859–883.

49.

Zhang

Yan

Peng

et al. (2014) Electrostatic pull-in instability in MEMS/NEMS: A review. Sensors and Actuators A-Physical 214: 187–218.