Nonlinear dynamical system of micro-cantilever under combined parametric and forcing excitations in MEMS

Sensors and Actuators A 119 (2005) 291–299

Nonlinear dynamical system of micro-cantilever under combined parametric and forcing excitations in MEMS Wenming Zhang∗ , Guang Meng State Key Laboratory of Vibration, Shock and Noise, Shanghai Jiao Tong University, 1954 Huashan Road, Shanghai 200030, PR China Received 3 March 2004; received in revised form 1 June 2004; accepted 28 September 2004 Available online 11 November 2004

Abstract This paper presents a simplified model for the purpose of studying the resonant responses and nonlinear dynamics of idealized electrostatically actuated micro-cantilever based devices in micro-electro-mechanical systems (MEMS). For the common cases of the micro-cantilever excited by periodic voltages, the underlying linearized dynamics are of a periodic or quasi-periodic system described by a modified nonlinear Mathieu equation. The harmonic balance (HB) method is applied to simulate the resonant amplitude frequency responses of the system under the combined parametric and forcing excitations. The resonance responses and nonlinearities of the system are studied under different parametric resonant conditions, applied voltages and various gaps between the capacitor plates. The possible effects of cubic nonlinear spring stiffness and nonlinear response resulting from the gas squeeze film damping on the system can be ignored are discussed in detail. The nonlinear dynamical behaviors are characterized using phase portrait and Poincare mapping in phase space, and the present analytical solutions and numerical simulations show that the nonlinear dynamical system is stable when choosing the rational parameters. This investigation provides an understanding of the nonlinear and chaotic characteristics of micro-cantilever based device in MEMS. © 2004 Elsevier B.V. All rights reserved. Keywords: MEMS; Micro-cantilever; Nonlinear; Parametric excitation; Forcing excitation

1. Introduction Most micro-electro-mechanical systems (MEMS) are inherently nonlinear, and the micro-scale effects and the coupled fields give rise to the complete nonlinearities in MEMS [1–3]. There exist intrinsic nonlinearities and exterior nonlinearities arising from coupling of different domains. In addition, mechanical nonlinearities, i.e., large deformations, surface contact, creep phenomena, timedependent masses and nonlinear damping effects, etc. also need to be considered [4–5]. Resonant sensors, which have the high sensitivity and resolution, low power consumption and digital output characteristics, have played an important role in MEMS [6]. Many micro-cantilever based MEMS ∗ Corresponding author. Tel.: +86 21 5474 4990x109; fax: +86 21 5474 7451. E-mail address: [email protected] (W. Zhang).

0924-4247/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.sna.2004.09.025

sensors, which build on the measurement of displacement and the smallest detectable motion, have been utilized for high precision chemical detection (e.g., MEMS nose) and small force detection. Some micro-cantilever mass sensors have been developed and applied as chemical sensors, biosensors and other sensors as well [7–10]. Many researchers have investigated the nonlinear dynamical behaviors of micro-cantilever based instrument in MEMS under various loading conditions. Passiana et al. [11] presented a micro-cantilever of an atomic force microscope (AFM) dynamic system to display the useful resonance behavior at kilohertz frequencies. Zook and Burns [12] calculated the natural frequencies of a micro-beam using the finite element method (FEM). But the results showed no differences between the beam model and the plate model due to the fact that their models excluded the electrostatic force and coupling. Choi and Lovell [13] computed the static deflection of a micro-beam numerically by the shooting method. The

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where (·) denotes the derivative with respect to t, y the vertical displacement of the micro-cantilever relative to the origin of the fixed plate, m the mass, k and c are the effective spring stiffness and damping coefficient of the simplified system, respectively. And according to parallel plate theory, the fringe effects at the edges of the plates are ignored [20,21], thus, FE , the electrostatic force between the capacitor plates (the fixed plate and the movable plate) generated by applying a voltage V(t), can be expressed as FE = Fig. 1. A simplified dynamical model of the micro-cantilever in MEMS.

conclusion accounted for both the electrostatic force and the model, whereas neglected the motion of the micro-beam under electrostatic effects. Ahn et al. [14] modeled a micro-beam under electrostatic actuation as a single degreeof-freedom spring-mass-damper system. The model assumed a linear spring and damping, thus neglected nonlinear response from the air damping. Serry et al. [15] considered the non-linearity of capacitor to investigate the Casimir effect in a MEMS model. Hung and Senturia [16] used an actuator beam model to determine minimum actuation voltage experimentally. Chan and Dutton [17] investigated a twodegree-of-freedom tilting model to describe the translation and rotation motions of capacitors. For a better understanding of those nonlinear characteristics, some valuable nonlinear dynamical analyses were presented [15]. Flores et al. [18] constructed a mathematical model of an idealized electrostatically actuated MEMS device for the purpose of studying the nonlinear dynamics of systems. Despite the roughly three decades of work in this area, the nonlinear dynamics of electrostatic systems remain relatively unexplored. The paper has the following structure. In Section 2, we describe the simplified mass-spring-damping model of an electrostatically actuated micro-cantilever based device in MEMS. In Section 3, we investigate the resonances of the nonlinear system under combined parametric and forcing excitations using the harmonic balance method (HB) and we study the nonlinear dynamic behaviors of the system under the effects of different applied voltages, cubic nonlinearity and nonlinear damping numerically and present the discussion in detail in Section 4. Finally, we end the paper in Section 5 with conclusions.

2. Dynamical model The electrostatically actuated micro-cantilever in MEMS is 4.5 um × 80 um × 200 um in dimensions [19], and the dynamical model is shown in Fig. 1. For this simplified massspring-damping system, the governing equation of motion for the dynamics of the system in MEMS is m¨y + cy˙ + ky = FE (t)

(1)

ε0 A V 2 (t) 2 (d − y)2

(2)

where ε0 is the absolute dielectric constant of vacuum (ε0 = 8.85 × 10−12 F/m), A the overlapping area between the two plates, and d is the gap between them. The reference state is m = 3.5 × 10−11 kg, k = 0.17 N/m, c = 1.78 × 10−6 kg/s, A = 1.6 × 10−9 m2 [19]. Introducing the following dimensionless variables
c k y ; τ = ω0 t; x = ; ζ = √ ; ω0 = d m km T =

ε0 A 2mω02 d 3

Therefore, Eqs. (1) and (2) give x¨ + ζ x˙ + x = T

V 2 (τ)

(3)

(1 − x)2

3. Resonance responses When a square root ac voltage signal V = [V0 cos(ωt)]1/2 is applied to the system, FE is a function of displacement y and time t. Since the force due to electrostatic interactions has a square dependence on the voltage applied, in order to isolate the parametric effects from harmonic effects, we can use a square rooted sinusoidal voltage signal [22]. 3.1. The case 1/(1 − x)2 ≈ 1 + 2x + 3x2 + O(ε3 ) For the case 1/(1 − x)2 ≈ 1 + 2x + 3x2 + O(ε3 ) when x 1, it should be noted that we are working with first and second order expansion and O(ε3 ) term is neglected here. The governing Eq. (3) can be rewritten as a modified nonlinear Mathieu equation x¨ + ψ(x, x˙ ) + x + f0 (x) + q(t)e(x) = p(t)

(4)

where ψ(x, x˙ ) = ζ x˙ ;

f0 (x) = 0;

e(x) = −2x − 3x2 ;

q(t) = TV0 cos t;

p(t) = TV0 cos Ωt;

Ω=

ω . ω0

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Introducing dimensionless variable a∗2 = 1 − Ω2 /Ω2 , and we assume the approximation solution for the system as

According to the solutions in Eq. (10), then Eq. (9) can be rewritten as  2 ∗ Ω a A0 + ΩζB0 − 23 TV0 a2 + 43 TV0 (B02 − A20 ) = TV0 Ω2 a∗ B0 − ΩζA0 − 23 TV0 A0 B0 = 0

x1 = a cos(Ωt + φ) + x10 = A0 cos Ωt + B0 sin Ωt −

1 TV0 Ω2

cos Ωt

We can find the relationship between the amplitude a, a2 = + B02 , and the dimensionless frequency a* from Eq. (11)

A20

e(x) ≈ e(A0 cos Ωt + B0 sin Ωt)

a∗2 =

=

1 Ce0 (A0 , B0 ) + 2

[Cer (A0 , B0 ) cos rΩt

r=1

+ Ser (A0 , B0 ) sin rΩt]

(6)

where 1 Cer (A0 , B0 ) = π



2π

e(A0 cos Ωτ + B0 sin Ωτ) cos rτ dτ

0

(7)

Ser (A0 , B0 ) =

1 π



2π

e(A0 cos Ωτ + B0 sin Ωτ) sin rτ dτ,

0

(r = 0, 1, 2, . . .)

(8)

The harmonic balance method is a simple, systematic approach and is not restricted to weakly nonlinear problems. Substituting Eqs. (5) and (6) into Eq. (4) and using the harmonic balance (HB) method [23], we yield 

(11)

(5)

For the further step of solution, we can neglect the small value x10 , then

∞ 

293

as T 2 V02 ((369/16)a6 +(189/16)a4 −6a2 −9)−3Ω2 ζ 2 a4 3Ω4 a4

(12)

Since a*2 ≥ 0, then a2 must satisfy the following inequality

6 + 189 a4 − 6a2 − 9 ≥ 3Ω2 ζ 2 a4 T 2 V02 369 a (13) 16 16 Therefore, if a2 satisfies with inequality (13), the nonlinear system will be stable under the combined parametric and forcing excitations. In addition, it is significant to study the nonlinear dynamic system in stable state. To illustrate the relationship between the square of amplitude a2 and that of frequency a*2 , we consider Ω = 1/2 and 1 the main conditions to be discussed. The resonant conditions relative to the different applied voltages V0 term in Eq. (12) is shown in Figs. 2 and 3. The relationships between the square of amplitude a2 and that of frequency a*2 are given for Ω = 1/2 and 1 resonance in MEMS. When the resonant parameter Ω varies from 1/2 to 1, the distribution of a2 changes accordingly and the value of a2 becomes larger for the same a*2 under different applied voltages (V0 = 10, 20, and 100 V). a2 is increasing gradually as a*2 increases, but decreases greatly with the increase the applied voltages V0 . In addition, when the applied voltages become larger, a2 tends

Ω2 a∗ A0 + Cf1 (A0 , B0 ) + C1 (A0 , B0 ) + 21 TV0 Ce0 (A0 , B0 ) + 21 TV0 Ce2 (A0 , B0 ) = TV0

Ω2 a∗ B0 + Sf1 (A0 , B0 ) + S1 (A0 , B0 ) + 21 TV0 Se2 (A0 , B0 ) = 0

(9)

where   1 2π   Cf1 (A0 , B0 ) = f0 (A0 cos τ + B0 sin τ) cos τ dτ = 0    π 0       1 2π    S (A , B ) = f0 (A0 cos τ + B0 sin τ) sin τ dτ = 0 0 f1 0   π 0        1 2π   C (A , B ) = (A0 cos τ + B0 sin τ, −ΩA0 sin τ + ΩB0 cos τ) cos τ dτ = ΩζB0  1 0 0 π 0    1 2π    S (A , B ) = (A0 cos τ + B0 sin τ, −ΩA0 sin τ + ΩB0 cos τ) sin τ dτ = −ΩζA0   1 0 0 π 0       Ce0 (A0 , B0 ) = −3(A20 + B02 )        Ce2 (A0 , B0 ) = − 23 (A20 − B02 )     Se2 (A0 , B0 ) = −3A0 B0

(10)

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Fig. 2. The relationship between the square of amplitude a2 and that of frequency a*2 of the micro-cantilever based device in MEMS when V0 = 10, 20, and 100 V with Ω = 1/2 and d = 1 um.

to a fixed value (e.g., a2 = 0.689 at V0 = 100 V). Therefore, it is very important to know what resonance exists in MEMS and the relationship between the amplitude and frequency. 3.2. The case 1/(1 − x)2 ≈ 1 + 2x + O(ε2 ) For the case 1/(1 − x)2 ≈ 1 + 2x + O(ε2 ) when x 1, notice that we are dealing with first order expansion and O(ε2 ) is neglected. e(x) term in modified nonlinear Mathieu Eq. (4) becomes e(x) = −2x, the governing equation gives x¨ + ζ x˙ + x − 2TV0 cos Ωt = p(t)

(14)

From Eq. (14), the relationship between the amplitude a and dimensionless frequency a* applying harmonic balance (HB) method yields 2   2 a∗ TV0 TV0 Ω a2 = (15) + Ω2 a∗2 + ζ 2 Ω3 a∗2 + Ωζ 2

Fig. 4. The relationship between the amplitude a and frequency a* of the micro-cantilever based device in MEMS when V0 = 5, 10, and 20 V with d = 1 um and Ω = 1/2.

the plates are illustrated in Figs. 4 and 5. With the increase of applied voltage (10, 20, and 100 V), the amplitude a changes obviously and the relationship between them varies correspondingly when the gap d and resonant parameter Ω satisfy d = 1 um and Ω = 1/2, respectively, as shown in Fig. 4. At the same time, as illustrated in Fig. 5, the amplitude a reduces quickly and tends to an approximation when the gap between the plates is becoming wider (e.g., a ≈ 0.005 at d = 5 um for Ω = 1 and V0 = 10 V). Therefore, the resonant parameter Ω, gap d and applied voltage V0 are the main effect factors on the measurement and detection of displacement of microcantilever based instrument in MEMS.

4. Numerical analysis and discussion 4.1. Effect of different applied voltages

The relationships between the amplitude a and frequency a* under different applied voltages and various gaps between

Introducing the vector [x1 , x2 ]T = [x, x˙ ]T , the state space representation of the micro-cantilever model resulting from governing Eq. (4) for the case 1/(1 − x)2 ≈ 1 + 2x + 3x2 +

Fig. 3. The relationship between the square of amplitude a2 and that of frequency a*2 of the micro-cantilever based device in MEMS when V0 = 10, 20, and 100 V with Ω = 1 and d = 1 um.

Fig. 5. The relationship between the amplitude a and frequency a* of the micro-cantilever based device in MEMS when d = 1, 3, and 5 um with V0 = 10 V and Ω = 1.

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Fig. 6. Phase portrait and Poincare mapping of the nonlinear dynamical system without considering the alternating current (ac) voltage and the nonlinear spring stiffness term when V0 = 20 V, Ω = 1/2 and d = 3 um. The simulation is scaled in time with relation t = ωt0 for numerical integration accuracy.

O(ε3 ) gives  x˙ 1 = x2 x˙ 2 = −ζx2 − x1 + (2x1 + 3x12 )TV0 cos Ωt + TV0 cos Ωt (16) Note that state Eq. (16) is a nonlinear, parametric and forcing excitations, time-varying and T-periodic model. The following parts focus on the simulating that we performed to validate the model. The phase portrait of a system is a representation of all its trajectories. Figs. 6 and 7 illustrate the phase portraits to investigate the nonlinear dynamical behaviors of the system. It can be found that the phase portraits, for Ω = 1/2 (see Fig. 6) and Ω = 1 (see Fig. 7), are two-dimensional close loops and their stable equilibrium is a fixed point A (in Poincare plane) for trajectories in phase space. Therefore, the system is stable for different resonant parameter without considering the

Fig. 7. Phase portrait and Poincare mapping of the nonlinear dynamical system without considering the alternating current (ac) voltage and the nonlinear spring stiffness term when V0 = 20 V, Ω = 1 and d = 3 um. The simulation is scaled in time with relation t = ωt0 for numerical integration accuracy.

295

Fig. 8. Phase portrait of the dynamical system without considering the nonlinear spring stiffness term when VP = 10 V, V0 = 5 V, Ω = 1/2 and d = 3 um. The simulation is scaled in time with relation t = ωt0 for numerical integration accuracy.

alternating current (ac) voltage and the nonlinear spring stiffness term. When the applied voltage, V, includes alternating current (ac) voltage, V0 (cos(ωt + φ), and polarization voltage, VP , V = [VP + V0 cos(ωt + φ) be considered for analyzing the nonlinearities of the system. Thus, the parametric excitation and forcing excitation terms in Eq. (4) should be rewritten as q(t) = T (VP + V0 cos Ωt)2 ; p(t) = T (VP + V0 cos Ωt)2

(17)

Phase portraits of the dynamical system under the effects of foregoing applied voltages are shown in Figs. 8 and 9. It is observed that the system still exists in stable motion with d = 3 um for Ω = 1/2 and 1. It is indicated that the motion of this system is a stable periodic one under different resonant conditions. From the results of the present harmonic balance

Fig. 9. Phase portrait of the dynamical system without considering the nonlinear spring stiffness term when VP = 10 V, V0 = 2 V, Ω = 1 and d = 3 um. The simulation is scaled in time with relation t = ωt0 for numerical integration accuracy.

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Fig. 10. Phase portrait (a) and Poincare mapping (b) of the micro-cantilever based device in MEMS when V0 = 20 V, VP = 4 V, Ω = 1, d = 1 um and k1 = 0.3. The simulation is scaled in time with relation t = ωt0 for numerical integration accuracy.

(HB) method, we can see that it gives good agreement with the analytical solutions. 4.2. Effect of cubic nonlinear spring Inducing the cubic nonlinear spring stiffness term k1 x3 (i.e., f0 (x) = k1 x3 ) and the system also runs under the applied voltage V = [VP + V0 cos(ωt + φ), thus the state space representation is given below   x˙ 1 = x2 

ner circle approaching a stable state. A Poincare mapping is a stroboscopic view of phase space [24]. The Poincare mapping section for the appearance of the three-dimensional trajectory when k1 = 0.3, Ω = 1 and d = 1 um is plotted in Fig. 10b. Since the Poincare mapping contains a finite number of fixed points, the motion of the micro-cantilever MEMS with cubic nonlinear spring is limit cyclic and stable. We assume, Ω = 1/2 to study the effect of resonant parameter Ω on the

(18) x˙ 2 = −ζx2 − (x1 + k1 x13 ) + (2x1 + 3x12 )T (V0 + VP )2 cos2 Ωt + T (V0 + VP )2 cos2 Ωt

The demonstration of chaotic motion under the effect of cubic nonlinear term helps us greatly to understand the nonlinear dynamics of a micro-cantilever based device in MEMS. In Fig. 10a, the phase portrait of the system (governing Eq. (17)) displays a three-dimensional strange attractor and the movement of its trajectory from the exterior circle to the in-

dynamical characteristic and stability of the system, and the phase portrait and Poincare mapping of the micro-cantilever MEMS when k1 = 0.3 and d = 1 um are shown in Fig. 11. We find that the Poincare section contains a closed curve and a finite number of fixed points. It is indicated that the motion

Fig. 11. Phase portrait (a) and Poincare mapping (b) of the micro-cantilever based device in MEMS when V0 = 20 V, VP = 4V, Ω = 1/2, d = 1 um and k1 = 0.3. The simulation is scaled in time with relation t = ωt0 for numerical integration accuracy.

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of the system lies between the limit cyclic motion and the quasi-periodic motion. Accordingly, it is important to select reasonable resonant conditions because certain resonant parameter causes chaotic behavior in the system with the cubic nonlinear spring.

Considering the effect of squeeze film damping on the system, ψ(x, x˙ ) term in Eq. (4) can be normalized and expressed as  k2 1 (21) ψ(x, x˙ ) = ζ + 2 3 x˙ d x

4.3. Effect of squeeze film damping

Substituting Eq. (21) into Eq. (4), it can be seen that the system holds two nonlinear terms, i.e., f0 (x) and Ψ (x, x˙ ). We study the nonlinearities and chaos of the system and analyze the stability characteristics under different conditions. Fig. 12 illustrates the micro-cantilever MEMS displacement trajectory, velocity trajectory, phase portrait and Poincare mapping. It is shown that the system behavior is close to that of a strange cyclic attractor and the motion of the system is not periodic. Since the phase portrait and Poincare mapping section (see Fig. 12c and d) contains close curves, the system corresponds to quasi-periodic motion. However, the displacement and velocity trajectories (see Fig. 12a and b) of the micro-cantilever MEMS display that the motion of the system is stable because the displacement and velocity become periodic after a few periods. It can be seen that the squeeze film damping will lead to instantaneous chaotic behavior in the micro-cantilever MEMS within some initial periods. Therefore, the effect of the squeeze film damping on the system can be neglected when the gap between the

MEMS devices are often characterized by structures that are a few microns in size, separated by micron-sized gaps. At these sizes, air damping dominates over other dissipation. Squeeze film damping may be used to represent the air damping experienced by the moving plates [25,26]. Starr [27] presented the expression of the damping force for a rectangular plate of dimensions 2W × 2L as Fsfd = −

16µcr W 3 L k2 y˙ = − 3 y˙ 3 y y

(19)

where µ is the viscous coefficient and W and L are the width and length of the micro-cantilever, respectively [19]. And cr is approximately equal to  W W cr = 1 − 0.6 0< <1 (20) L L

Fig. 12. Simulation of the micro-cantilever MEMS behavior: (a) displacement trajectory, (b) velocity trajectory, (c) phase portrait, (d) Poincare mapping. The parameters used in the simulation are: the alternating current (ac) voltage V0 = 20 V; the polarization voltage VP = 4V; the resonant parameter Ω = 1; the gap d = 1 um; the cubic stiffness coefficient k1 = 0.3; the scale parameter of the squeeze film damping k2 = 2.2812 × 1010 Pa um4 s; and the initial displacement x0 = 0 and the initial velocity x˙ 0 = 0.

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plates is very small and the system becomes stable after a few periods. 4.4. Discussion According to the analytical and numerical solutions of the nonlinear governing equation of motion, the effective spring stiffness can be regarded as linear when the displacement of the micro-cantilever is very small (see Eq. (16) and Figs. 6–9. In addition, the nonlinear spring stiffness term k1 x3 weakly affects the whole system under the combined parametric and forcing excitations, particularly when the gap between the two plates is becoming larger. This dynamical system still has periodic solution with some resonant conditions (see Fig. 10). It should be noted that a different resonant parameter can leads the system to a quasi-periodic motion state with cubic nonlinear spring (see Fig. 11). Although the gap between the two plates can be assumed to very small, the interaction forces, which include van der Waals force, FV , and Casimir force, FC except for electrostatic force, cannot be neglected when the gap is reaching a certain micro-magnitude [28–30]. Furthermore, the effects from both of these interaction forces will give rise to adhesion between the micro-cantilever and fixed plate. These nonlinear phenomena should be considered in the designs and fabrications of micro-cantilever based instrument in MEMS. In micro-scale, the nonlinear air damping resulting from the hydrokinetic plays a very important role in MEMS. If the deformation of micro-cantilever based instrument in MEMS is larger under the outside excitations, the effects of nonlinear damping cannot be neglected. Nonlinear damping (intrinsic structure nonlinearities and exterior nonlinearities) can lead the system to chaotic states. The displacement of a micro-cantilever based device in MEMS, which is used for measurements and detections, is assumed to be very small corresponding to the gap between the plates. The squeeze film damping causes instantaneous chaotic behavior in the micro-cantilever MEMS within some initial periods (see Fig. 12). Therefore, it is rational to neglect the effect of the squeeze film damping on the system when the gap between the plates is very small in this paper. Furthermore, the relationship between the parametric excitation and forcing excitation are correlative such that the parametric excitation term mainly comes from the forcing excitation term in the system (demonstrated in governing Eqs. (4), (16) and (18)). Accordingly, we can change the parametric resonant states by tuning the exterior forcing excitation depending on the micro-measurement and micro-detection in MEMS.

5. Conclusions The nonlinear behaviors in certain conditions of micro-cantilever based device in MEMS are investigated analytically and numerically. The relationships between the amplitude and frequency under different applied voltages

and various gaps between the plates are determined. Resonance responses of the system under combined parametric and forcing excitations are studied in detail. It is found that the resonant parameter Ω, gap d and applied voltage V are the main effect factors on the measurement and detection of the displacement of micro-cantilever based instrument in MEMS. The effects of different applied voltages, cubic nonlinear spring and squeeze film damping on the nonlinear and chaotic behaviors of the system are discussed. It is obtained that the motion of the system with the cubic nonlinear spring for Ω = 1/2 and squeeze film damping is quasi-periodic but stable. The results provide a possible reference for the choice of reasonable resonant conditions, design and industrial applications of such micro-cantilever based MEMS devices.

Acknowledgements The authors are grateful to Drs. X.L. Leng and C.X. Zhao. This project is supported by National Outstanding Youth Foundation of People’s Republic of China (no.10325209).

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Biographies Wenming Zhang received the BS degree in mechanical engineering in 2000 and MS degree in mechanical design and theories in 2003, both from Southern Yangtze University. He is currently working toward his PhD at State Key Laboratory of Vibration, Shock & Noise in Shanghai Jiao Tong University. His research interests include micro-motor, vibration analysis, nonlinear dynamics and control in MEMS. Guang Meng received the PhD in vibration engineering from the Northwestern Polytechnical University in 1988 and worked from 1989 to 1993 in Texas A&M University, Berlin Technical University and New South Wales University as Research Assistant, Alexander von Humboldt Fellow and ARC Fellow. He is the “Cheung Kong” Chair Professor and Deputy Dean of School of Mechanical and Power Engineering, Shanghai Jiao Tong University. His research interests include vibration analysis and control in MEMS, rotor dynamics, state monitoring and fault diagnosis, smart structure and smart control and nonlinear vibration.