Bifurcation behavior for mass detection in nonlinear electrostatically coupled resonators

Journal Pre-proof Bifurcation behavior for mass detection in nonlinear electrostatically coupled resonators Lei Li, Wenming Zhang, Jing Wang, Kaiming Hu, Bo Peng, Mingyu Shao

PII: DOI: Reference:

S0020-7462(19)30608-0 https://doi.org/10.1016/j.ijnonlinmec.2019.103366 NLM 103366

To appear in:

International Journal of Non-Linear Mechanics

Received date : 3 September 2019 Revised date : 22 October 2019 Accepted date : 26 November 2019 Please cite this article as: L. Li, W. Zhang, J. Wang et al., Bifurcation behavior for mass detection in nonlinear electrostatically coupled resonators, International Journal of Non-Linear Mechanics (2019), doi: https://doi.org/10.1016/j.ijnonlinmec.2019.103366. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd.

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Bifurcation behavior for mass detection in nonlinear electrostatically coupled resonators Lei Li1,2, Wenming Zhang1*,Jing Wang3, Kaiming Hu1, Bo Peng1, Mingyu Shao2 1

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao

Tong University, Shanghai 200240, China School of Transportation and Vehicle Engineering, Shandong University of Technology, Zibo 255049, China

3

Library, Tianjin University, Tianjin, 300350, China

of

2

Abstract：The nonlinear coupled vibrations widely exist in coupled resonant structures, which can lead to complex dynamic bifurcation behavior and expand the research scope of fundamental

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physics. A new micro-mass detection method is proposed by using bifurcation jumping phenomenon in nonlinear electrostatically coupled resonators in this article. Considering the fundamental frequency excitation, the one-to-one internal resonance equations to describe electrostatically coupled resonant sensor are obtained by using Hamilton’s principle and Galerkin method. Then, the perturbation analysis method is introduced to study the response and stability of

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the system for small amplitude vibration. Through bifurcation analysis, it is found that the isolated response branches appear in nonlinear electrostatically coupled resonators and present the physical conditions of this phenomenon. Typically, we demonstrate the exploitation of the bifurcation jump phenomena of two electrostatically coupled microbeam resonators to realize the mass quantitative

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detection and threshold detection, which overcomes the detection inaccuracy caused by frequency drift in the nonlinear vibration. Finally, the numerical experiments verify the validity of the method. The results of this paper can be potentially useful in micro-mass detection. Key word: Bifurcation; MEMS; Coupled resonance; mass detection

1. Introduction Electrostatically

coupled

resonators

have

been

gradually

applied

in

many

Micro-electro-mechanical-systems (MEMS) sensors [1-3]. Due to their great potential and unique characteristics, coupling resonant sensors have the advantages of small, fast, high sensitivity with low noise-sensitivity. Meanwhile, they require only a small amount of power to operate [4, 5]. However, the existence of structure nonlinearity and nonlinear electrostatic force can exhibit

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complex nonlinear dynamic behaviors [6-9], which seriously affect the detection performance of micromass sensor. For example, nonlinearities can cause shifts in their resonant frequency and lead to distortion of the measured results [10]. Meanwhile, the nonlinear coupled vibrations can also improve resonator performance and expand the research scope of fundamental physics [11]. Recently, bifurcation jumping behaviors were exploited to improve the sensitivity of micro mass detection [12, 13]. Younis et al [14] utilized dynamic instabilities and bifurcations in single degree of freedom MEMS system to realize novel methods and functionalities for mass sensing. In this 1

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paper, coupled resonant structures are considered to study the complex bifurcation behaviors and their physical conditions. Furthermore, we try to use the bifurcation jumping phenomenon of two-degree-of-freedom system to complete the detection of mass, which can improve the sensitivity of the sensor. The nonlinear coupled vibrations widely exist in multi-degree of freedom resonant structure, which introduces rich nonlinear phenomena into the MEMS research [15-17]. Yu et al. [18] utilized 1:2

of

internal resonances in two distinct doubly clamped micromechanical beam resonators to realize the frequency stabilization. Due to the strong coupling between the two flexural modes, the frequency

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stabilization can be achieved over a much broader range of operation. Ouakad et al. [19] studied the one-to-one and three-to-one internal resonances caused by the nonlinear modal interaction of MEMS actuator. The results showed that the energy can be exchanged between the successive modes of the structure for some values of initial rise. Considering electrostatic actuation at forced and internal resonance frequencies, Atabak et al. [20] used the analytical and experimental method

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to investigate the nonlinear mode coupling in a tuning fork microresonator. Houri et al. [21] studied the dependence of bifurcation behaviors on the oscillator’s amplitude and frequency and reproduced the observed behavior using a system of nonlinearly coupled equations which show interesting scaling behavior. Tchakui et al. [22] designed a ring of three unidirectionally coupled nonlinear

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MEMS and studied the complex bifurcation behaviors when the coupling coefficient varies. Considering coupling vibrations among the microbeam modes involving the first mode, Younis et al [23] investigated the one-to-three internal resonance between the first and second modes. It is noted that the second mode cannot obtain energy from the first mode. Besides, the coupled microcantilever array was also studied. Kambali et al. [24] revealed an intricate structure for small damping that includes both quasiperiodic and nonstationary chaotic-like energy transfer between the elements of the array. Meanwhile, recent experimental work was also used to study the nonlinear coupled vibrations between different eigenmodes of coupled mechanical resonators [25, 26]. In a word, the coupled vibration behaviors have attracted more and more attention. These studies are very important to reveal the mechanism of the complex dynamic bifurcations and improve oscillator performance [27].

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Recently, nonlinear coupled vibrations have been gradually introduced into the MEMS mass sensors. Mass measurements based on the eigenmode shift of coupled resonance structures provide much higher sensitivity than the method based on the natural frequency shift of a single microbeam. However, the nonlinearity in flexural MEMS results dependence of resonant frequency with amplitude, known as the amplitude-frequency effect [28]. Many MEMS mass sensors depend on operating in a resonant mode at a particular frequency [29]. Here, nonlinearities can cause shifts in their resonant frequency and lead to distortion of the measured results. Recently, efforts to exploit 2

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nonlinearities for improved performance have emerged and are also described. Du et al. [30] exploited the internal resonance of coupled ductile cantilevers to realize a new design concept for mass sensors. The frequency enhancement and amplitude amplification mechanism of internal resonance oscillators revealed the potential application of sensors. Pandit et al. [31] used the energy localization within weakly coupled nonlinear MEMS resonators to realize detection, which greatly improved the resolution of the detection. Kasai et al. [32] proposed the concept of virtual cantilever

of

beam and improved the sensitivity of resonant sensor by using a virtual cantilever virtually coupled with a real cantilever. Besides, some mass sensors based on bifurcation behaviors were proposed

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[14]. The techniques are much less susceptible to damping than frequency shift-based approaches because bifurcations may be activated regardless of damping so long as the excitation level exceeds a critical threshold [33]. Kumar et al. [34] provided a mass sensing method relying on jumps at limit points of the nonlinear frequency response. Zhang et al. [35] and Thomas et al. [36] studied the parametric resonance and showed that the sensitivity is highly increased due to the sharpness of

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amplitude transition in this regime. Similarly, Nguyen et al. [37] also proposed a mass threshold detection method based on bifurcation jumping characteristics. Results showed that sudden jumps in amplitude made the detection of a very small mass possible. Hasan et al. [38] studied the intelligent adjustable threshold pressure switch. When the pressure exceeds the critical threshold,

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the system can be induced to produce amplitude jump, realizing the rapid sensing of pressure value. Recent bifurcation-based mass sensing studies have employed directly excited nonlinear oscillators to induce critical jump events [10]. However, most mass sensors based on bifurcation are designed for single degree of freedom systems. There are few researches on mass threshold detection by using nonlinear coupled vibration behavior in Multi-degree of freedom system. It can be concluded from the above research status that the complex nonlinear dynamical behaviors are very important in the design of MEMS mass sensors and should be taken into account [39]. Coupling resonant sensors have the advantages of small, fast, high sensitivity with low noise-sensitivity. However, coupled resonant structures can lead to mode coupling behaviors, nonlinear dynamic behaviors and various bifurcation mechanisms, which greatly increases the difficulty of the analysis. Besides, the resonance frequency of the coupling resonator is affected by

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the driving voltage, which can affect mass test results. The mass sensor based on bifurcation behavior can realize high precision detection by using nonlinear vibration. Meanwhile, multi-degree of freedom MEMS mass sensor is the trend of future development. There are few studies on the influence of bifurcation behavior on mass detection in multiple degrees of freedom system. Here, we propose a simplified coupled capacitance resonant structure and study the complex bifurcation behaviors in nonlinear electrostatically coupled resonators. The physical conditions of different bifurcation mechanisms are presented. Typically, we demonstrate the exploitation of the bifurcation 3

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jumping behavior of two electrically coupled microbeam resonators to realize the mass detection, which overcomes the detection inaccuracy caused by frequency drift in the nonlinear vibration. The work of this article may improve the performance of micro mass sensors. The rest of this paper is organized as follows. In Sect. 2, two degrees of freedom equations are obtained by the Hamilton’s principle and Galerkin discretization. In Sect. 3, we introduce the perturbation method to produce an approximate solution. In Sect. 4, complex bifurcation behaviors

of

are analyzed in nonlinear electrostatically coupled resonant sensors. In Sect. 5, we present a new mass detection method with the bifurcation jumping phenomenon. Numerical experiments verify

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the validity of the theory. Finally, summary and conclusions are presented in the last section.

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2. Problem formulation

Fig. 1 A schematic diagram of two electrically coupled microbeam resonators

Fig. 2 A schematic diagram of added mass in microbeam 1

Multi-degree of freedom coupled resonant structures are gradually widely used in MEMS mass sensors [1]. Conventionally MEMS mass sensors use only a single vibration mode to realize detection, and the output signal of the sensor is typically a frequency shift caused by the added mass. Compared to using a single vibration mode to realize detection, the sensitivity of multi-degree of freedom coupled resonant structures can be improved by several orders of magnitude. Besides, the mode coupled vibration behavior exists in the resonant structure, which can lead to complex

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dynamic bifurcation behavior and affect the performance of the device. In this paper, we try to study the complex bifurcation behaviors in mode coupled vibration. Besides, we use bifurcation jumping phenomenon to realize the mass detection. A typical capacitive coupling structure was proposed to study the dynamic performance of nonlinear system [40]. Fig.1 shows a simplified model of two degrees of freedom MEMS resonator which plays an important role in the realization of mass sensor. By applying respective electrodes, the resonator 1 and resonator 2 can be separately actuated. Here, 4

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the resonator 1 is a clamped-clamped beam driven by means of a bias voltage and an AC voltage component, and the clamped-clamped beam 2 is driven by the coupling capacitor. Adsorption material is added to microbeam 1. Then, a lumped mass m is added at x  L1 , as shown in Fig. 2. Additional mass can be detected by investigating the dynamic bifurcation behaviors of the device. Consindering Hamilton’s principle, the equations of motion that govern the transverse deflections

of

wˆ 1 ( xˆ , tˆ ) and wˆ 2 ( xˆ , tˆ ) can be obtained

 2 wˆ1 wˆ  EI1wˆ 1iv  c 1 2 ˆ t ˆt ˆ tˆ)]2  b[V  Vac cos(  0bVdc2 EA L  ( 1  wˆ12 dx ) wˆ1  0 dc  2L 0 2(d  wˆ1 ) 2 2(d  wˆ 1  wˆ 2 ) 2

 A2

 2 wˆ 2 wˆ EA  EI 2 wˆ 2iv  c 2  ( 2 2 ˆ ˆ t t 2L



L

0

pro

[  A1   ( x  L1 )m]

wˆ 22 dx) wˆ 2 

 0bVdc2

2(d  wˆ 2  wˆ1 )2



 0bVdc2

2(d  wˆ 2 )2

(1)

(2)

The boundary conditions of the microbeam 1 and microbeam 2 are as follows

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wˆ1(0, tˆ)  wˆ1 (0, tˆ)  wˆ1 ( L, tˆ)  wˆ1( L, tˆ)  0

wˆ 2 (0, tˆ)  wˆ 2 (0, tˆ)  wˆ 2 ( L, tˆ)  wˆ 2 ( L, tˆ)  0 where wˆ i 

wˆ i for i =1, 2. xˆ

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It should be noted that the coupling term of the system comes from the electrostatic force, and the squeeze film force between the beams is not included [41, 42]. The internal resonance of coupled resonance structure can improve the detection accuracy of micro sensor [1]. In order to achieve the internal resonance, the beams of slightly different thicknesses are introduced to realize two close but different frequencies of the system, which can be potentially useful in mass sensor. In Eqs. (1)(2), A1=bh1 and A2=bh2 represent the cross section areas of microbeam 1 and microbeam 2. Besides, xˆ is the position along the beam length, I1 and I2 are moments of inertia of the cross sections, L is

the length of beam, E is Young’s modulus, tˆ is the time, ρ is the material density, b is the microbeam width, d is the gap width, and  0 is the dielectric constant of the gap medium. The last two terms in Eqs. (1) and (2) represent the parallel-plate electric actuation.

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Then, the non-dimensional variables are introduced

w1 

wˆ 1 wˆ xˆ EI1 , w2  2 , x  , t  tˆ L d d  A1 L4

Substituting the non-dimensional variables into Eqs. (1)- (2), yields the following non-dimensional equations of motion of the coupled resonant structure

5

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1  2 w1  2 w1 w  2  w1iv  cn 1  (1  w12 dx) w1 2 0 t t t 2 2 2VdcVac cos t  (Vac cos t )2 Vdc Vdc  2     2 2 (1  w1 ) 2 (1  w1  w2 ) 2 (1  w1 )2

(3)

1 Vdc2 Vdc2  2 w2 w2 2 iv         w c w dx w +  (  )   2 2 1 0 2 2 2 2 n t 2 t (1  w2  w1 )2 (1  w2 ) 2

(4)

1

of

with the following boundary conditions

w1 (0, t )  w1(0, t )  w1 (1, t )  w1(1, t )  0

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w2 (0, t )  w2 (0, t )  w2 (1, t )  w2 (1, t )  0 The parameters appearing in Eqs. (3)- (4) are

d h1

1  6  ( ) 2 ,  2 

6 0 L4  ( x  L1 L)m h2 2 6 0 L4      ( )   , , , 1 2 2  A1L h1 Ed 3h13 Ed 3h12 h2

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where h1 and h2 represent thicknesses of microbeam 1 and microbeam 2, 1 represents additional equivalent mass, 2 represents error coefficient between microbeam 1 and microbeam 2. Li et al. [40] found that the initial point of the system was stable when the DC voltage was small. Then, the electrodynamic terms of Eqs. (3)- (4) is expanded, and we obtain

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1  2 w1  2 w1 w  2  w1iv  cn 1  (1  w12 dx ) w1 2 0 t t t 2 2   2Vdc (4 w1  2 w2  3w2  6 w1w2  8w13  4 w23  12 w1w22  12 w12 w2 )  2 2VdcVac cos t

1

1  2 w2 w   2 w2iv  cn 2  (1  w22 dx ) w2  2 0 t t 2 2  2Vdc (4 w2  2 w1  3w1  6 w1w2  8w23  4 w13  12 w1w22  12 w12 w2 )

(5)

(6)

Due to Vdc  Vac [40], (Vdc  Vac cos t )2  Vdc2  2VdcVac cos t is obtained. The

solutions 

of

Eqs.

(5)-

(6)

can

be



expressed

as

w1 ( x, t )   u1,i (t )1,i ( x) and i 1

w2 ( x, t )   u2,i (t )2,i ( x) , where 1,i and 2,i are the i-th linear undamped modes shape of the i 1

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microbeam 1 and micorbeam 2. Then, the linear undamped eigenvalue equations are obtained

1,ivi  1,2i1,i  4 2Vdc21,i 1,i (0)  1,i (1)  1, i (0)  1, i (1)  0

22,ivi   2,2 i2,i  4 2Vdc22,i 2,i (0)  2,i (1)  2, i (0)  2, i (1)  0

(7)

(8)

Substituting Eqs. (7)- (8) into the resulting Eqs. (5)- (6), multiplying by 1,i , 2,i , and integrating the 6

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outcome from x=0 to 1, yield

(1   )

d 2u1,n dt

 cn

2

du1,n dt

M

 1,2nu1,n  2 2Vdc2  u2,i  2,i1, n dx 

M

1

0

i 1

M

 [      dx     1

1

1 0 1,i 1, j

i , j , k 1

0 1, k 1, n

dx

8 2Vdc2  1,i1, j1, k1, n dx]u1,i u1, j u1,k  3 2Vdc2  u2,i u2, j  2,i2, j1,n dx 1

0

1

0

i , j 1

M

M

6 2Vdc2  u1,i u2, j  1,i2, j1,n dx  4 2Vdc2  u2,i u2, j u2, k  2,i2, j2,k1,n dx 1

0

M

(9)

0

i , j 1

of

i , j 1

1

M

12 2Vdc2  u1,i u2, j u2,k  1,i2, j2, k1, n dx  12 2Vdc2  u1,i u1, j u2,k  1,i1, j2, k1, n dx 1

0

i , j 1

1

0

i , j 1

 2 2VdcVac cos t  1, n dx 0

d 2u2,n dt

2

 cn

du2,n dt

M

pro

1

  2,2 nu2, n  2 2Vdc2  u1,i  1,i2,n dx  i 1

1

0

M

M

 [     

i , j , k 1

1

1 0

2,i 2, j

1

dx  2, k2,n dx 0

8 2Vdc2  2,i2, j2, k2, n dx]u2,i u2, j u2,k  3 2Vdc2  u1,i u1, j  1,i1, j2,n dx 1

0

1

0

i , j 1

M

M

6 2Vdc2  u1,i u2, j  1,i2, j2, n dx  4 2Vdc2  u1,i u1, j u1, k  1,i1, j1,k2,n dx i , j 1

0

M

1

re-

1

0

i , j 1

M

(10)

12 2Vdc2  u1,i u2, j u2, k  1,i2, j2,k2, n dx  12 2Vdc2  u1,i u1, j u2, k  1,i1, j2,k2, n dx 1

0

i , j 1

0

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0

i , j 1

1

where  = 1,2i ( L1 L)m  A1 L .

In this paper, we take the geometric and the material parameters as E  169Gpa ,   2300 kg m 3 ,

L  150μm , h1  1μm , d  1.5μm and b  10μm [40]. Firstly, we introduce the software COMSOL to obtain the Finite element results by using the Multi-field solver [43]. Fig. 3 shows the first four vibrations modes of the electrostatic coupled resonant structure. It is found that the first vibration mode and the third vibration mode are out-of-phase. The second vibration mode and the

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fourth vibration mode are in-phase.

Fig. 3 The first four modes of the MEMS sensor in COMSOL.

Besides, it is discovered that the natural frequency of the first mode is close to that of the second 7

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mode. This article mainly investigates the complex bifurcation behaviors under the fundamental frequency. Thus, we take M=1 and obtain equations of motion of two degrees of freedom (1   )

1 1 1 d 2u1 du  cn 1  12u1  2 2Vdc2  21dxu2  [1  12 dx  11dx 2 0 0 0 dt dt 1

1

1

0

0

0

8 2Vdc2  14 dx]u13  3 2Vdc2 u22  221dx  6 2Vdc2 u1u2  122 dx

   dx  12 V u u    dx u u    dx  2 V V   dx cos t

4 V u 12 2Vdc2

1 2 2 2 2 2 dc 1 2 0 2 1

1 2 3 1 2 0 1 2

1

2 dc ac 0 1

(11)

of

1 2 3 3 2 dc 2 0 2 1

1

pro

1 1 1 d 2u2 du  cn 2   22u2  2 2Vdc2  21dxu1  [1  22 dx  22 dx 2 0 0 0 dt dt 1

1

8 2Vdc2  24 dx]u23  3 2Vdc2 u12  122 dx  6 2Vdc2 u1u2  221dx 0

0

1 2 3 3 2 dc 1 0 1 2

   dx  12 V u u    dx  0

4 V u

uu

(12)

   dx

1 2 2 2 1 2 0 1 2

re-

12 2Vdc2

0

1 2 2 3 2 dc 1 2 0 2 1

To calculate the natural frequencies of the system, the linear Jacobi matrix is obtained

2 2Vdc2  21dx  0   1+     22 

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 12  1+  J  1  2 2Vdc2  21dx 0 

1

(13)

Then, the resonant frequencies 1 and 2 can be obtained by solving the eigenvalues of the Eq. (13).

4  V 4 1 2 (  22   2 )( 1   2 )  2 2 dc (  21dx) 2  0 0 1+  1+ 

(14)

Fig. 4 shows the variation of the first natural frequency and the second natural frequency versus the size and position of the additional mass when Vdc  4V . The increase of the additional mass can increase the equivalent mass of the system. Thus, there is a negative correlation between the natural frequency and the additional mass of the microbeam 1. It is worth noting that the additional mass has an important effect on the first natural frequency. However, it has little effect on the second

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natural frequency. The closer the additional mass is to the middle of the microbeam, the greater the equivalent mass of the system. The finite element results obtained by the COMSOL are introduced to verify the theoretical results. Besides, the effect of DC voltage on the first and second natural frequencies is also studied, as shown in Fig. 5. As the voltage increases, the first natural frequency decreases sharply and approaches 0. Here, the certain critical value means that the Jacobian matrix is zero. Through finite element analysis, when the DC voltage is more than a certain critical value, the pull-in occurs. Meanwhile, the critical value is also called the pull-in voltage. From Fig. 5, it is 8

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found that the added mass can not affect the pull-in voltage of the system. When the driving voltage is small, the first natural frequency is approximately equal to the second natural frequency, and hence we study the one-to-one internal resonance when the microbeam 1 is driven by fundamental

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frequency.

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Fig. 4 Variation of the first natural frequency and the second natural frequency versus the size and position of the additional mass (the lines denote the theoretical results and the points denote the COMSOL results)

Fig. 5 Variation of the first natural frequency and the second natural frequency of coupled resonators with various values of the DC voltages (the lines denote the theoretical results and the points denote the COMSOL results; Black represents the results without considering the added mass and red represents the results with considering the added mass )

To make it easy to express, we use dimensionless coefficients to represent the integral terms and rewrite Eqs. (11)- (12) as

d 2u1 du  cn 1  12u1  k1l u2  k1au13  k1bu22  k1cu1u2  k1d u23  k1eu1u22  k1 f u12u2  f cos t (15) 2 dt dt

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(1   )

d 2u2 du  cn 2   22u2  k2l u1  k2 au23  k2bu12  k2cu1u2  k2 d u13  k2eu2u12  k2 f u22u1  0 2 dt dt

(16)

Where

9

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1

1

1

1

0

0

0

0

k1l  2 2Vdc2  21dx, k1a  1  12 dx  11dx  8 2Vdc2  14 dx 1

1

1

0

0

0

k1b  3 2Vdc2  221dx, k1c  6 2Vdc2  122 dx, k1d  4 2Vdc2  231dx 1

1

1

0

0

0

k1e  12 2Vdc2  2212 dx, k1 f  12 2Vdc2  132 dx, f  2 2VdcVac  1dx 1

1

1

1

0

0

0

0

k2l  2 2Vdc2  21dx, k2 a  1  22 dx  22 dx  8 2Vdc2  24 dx

   dx, k  6 V    dx, k  12 V    dx, k  12 V    dx

k2b  3 V k2 e

1 2 2 2 dc 0 2 1

2c

1 2 2 2 2 dc 0 2 1

2f

1

2d

 4 2Vdc2  132 dx 0

1 2 3 2 dc 0 2 1

of

1 2 2 2 dc 0 1 2

pro

The parameters that appear in Eq. (15)-(16) can be obtained by using the mode function, geometric dimensions and driving voltage. From Eqs. (15)- (16), we find that resonator 1 is driven by harmonic electrostatic force, and then the vibration energy is transferred from the resonator 1 to the resonator 2 by coupled capacitance. When the driving frequency is near the natural frequency, the coupled vibration can occur. Additional mass can change the natural frequency of resonator 1 and dynamic behavior of the system.

3 Perturbation analysis

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affect the transfer of vibration energy. Then, the additional mass can be detected by observing the

urn al P

In this section, the perturbation analysis method is introduced to study the coupled vibration behavior of nonlinear systems with two degrees of freedom. In section 2, it is found that the first natural frequency of the system is close to the second natural frequency. 1:1 internal resonance under the fundamental frequency excitation is considered. When the vibration amplitude is relatively small, the response frequencies of the resonator 1 and the resonator 2 are equal to the excitation frequency. Then, we utilize the method of multiple scales to investigate the response of the coupled resonant sensor. To indicate the significance of each term in the Eqs. (15)- (16), ε is introduced as a small bookkeeping parameter. To obtain the approximate solution, the electrostatic force term f  O( 3 ) , u1  O( ) , u2  O( ) are considered. Besides, the magnitude of the coupled terms and dissipative terms should be consistent with that of the external excitation. We can rewrite the Eqs. (15)- (16) as [40]:

d 2u1 du   2cn 1  2u1   ( 12  2 )u1   2 k1l u2  k1a u13  k1bu22  k1cu1u2  k1d u23 2 dt dt 2 2 3  k1eu1u2  k1 f u1 u2   f cos t

(17)

d 2u2 du   2 cn 2  2u2   (  22  2 )u2   2 k2l u1  k2 a u23  k2bu12  k2 cu1u2 2 dt dt 3  k2 d u1  k2eu2u12  k2 f u22u1  0

(18)

Jo

(1   2 )

In order to establish the relationship between the natural frequencies of resonators and the driving 10

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frequency, detuning parameters  and  are introduced and defined by

1  2  ,   1  

(19)

The solutions of Eqs. (17)- (18) are written as u1   u11 (T0 , T1 , T2 )   2u12 (T0 , T1 , T2 )   3u13 (T0 , T1 , T2 )

where Tn   n t .

of

u2   u21 (T0 , T1 , T2 )   2u22 (T0 , T1 , T2 )   3u23 (T0 , T1 , T2 )

(20)

Substituting Eqs. (19)- (20) into Eqs. (17)- (18) and equating coefficients of like powers of ε yield O   1  : D02u11   2u11  0

pro

D02u21   2u21  0

2 O   2  : D02u12   2u12  2 D0 D1u11  k1bu21  k1cu11u21

D02u22   2u22  2 D0 D1u21  k2bu112  k2 c u11u21

(21)

(22)

O   3  : D02u13   2u13  2 D0 D2u11  D12u11  2 D0 D1u12  cn D0u11  k1l u21

re-

3 2  k1a u113  k1d u21  k1eu11u21  k1 f u112 u21  2k1b u21u22

 k1c u11u22  k1c u12u21  ( 2  12 )u11    2u11  f cos t D02u23   2u23  2 D0 D2u21  D12u21  2 D0 D1u22  cn D0u21  k2l u11

(23)

3 2  k2 a u21  k2 d u113  k2 eu21u112  k2 f u21 u11  2k2b u11u12

urn al P

 k2 c u11u22  k2 c u12u21  ( 2   22 )u21

The general solutions of Eqs. (21) can be written as

u11 (T0 , T1 , T2 )  A11 (T1 , T2 )eiT0  A11 (T1 , T2 )e iT0 u21 (T0 , T1 , T2 )  A21 (T1 , T2 )eiT0  A21 (T1 , T2 )e  iT0

(24)

It is convenient to express A11 and A21 in the polar form

A11 

1 i 1 a1e , A21  a2 ei 2 2

where a1 and a2 indicate the amplitudes of the microbeam 1 and microbeam 2, respectively. Substituting Eq. (24) into Eqs. (22)- (23), yields the secular terms

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c  da1    f  sin   n a1  ( 1 a2  2 a12 a2  3 a23 )sin(   )  4 a1a22 sin(2  2 ) 2 2     dt  1 4 3 5   1 2 2 2 d f cos       a1  a2  ( a2  a2  a1a2 ) cos(   )  6 a22 cos(2  2 ) 2a1 2     dt a1 a1

    c da2   n a2  ( 5 a1  6 a1a22  7 a13 ) sin(   )  8 a2 a12 sin(2  2 )  2    dt 7 2 8 2 5 10 3 11  d  (    )  a2  a1  ( a1  a1  a1a2 ) cos(   )  12 a12 cos(2  2 )     dt a2  a2 

(25)

where 11

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k1 f 7 k1b k2b k1c k2 c k1l 3k 5k k ; 2   ;  3   1d  1b 12c   2 2 2 8 12 24 8 24 k2 f 7 k1b k2b k1c k2 c k1e k1b k2 c k12c k2l ;5   ;6   4       8 2 8 4 2 8 2 12 2 24 2 k k k k2 3k 5k k  7   2 d  2b 22c ;  8   2e  2b 12c  2 c2 8 24 8 4 8 k1e k1b k2 c k2 3k1a 5k2b k1c 1      1c 2 ; 2   2 2 4 6 12 8 24 3 k 3k 5k k 5k k 5k k k k k k2 4   1d  1b 12c ; 5   1 f  1b 22b  1c 22c ; 6   1e  1b 22c  1c2 8 8 8 24 12 24 4 8 k2 e k2b k1c k22c 3k2 a 5k1b k2 c 7   ; 8      8 24 2 4 6 2 12 2 3k 3k 5k k 5k k 5k k k k k k2 10   2 d  2b 22c ; 11   2 f  1b 22b  1c 22c ; 12   2e  2b 12c  2 c2 8 24 8 12 24 8 4 8

pro

of

1  

The stability of the periodic solutions can be determined by the method in Ref. [8].

4. Complex bifurcation behavior

In this section, the complex dynamic bifurcation behaviors of the electrostatic coupled resonant

re-

sensor in the case of weak coupling are considered. Besides, we try to deduce the physical conditions of different bifurcation mechanisms. When the DC voltage is small, it is found that the nonlinear coupling terms have little impact on the system. Then, we can rewrite the bifurcation Eq. (25) as

urn al P

c  da1 f  sin   n a1  1 a2 sin(   ) dt 2 2      d f  cos   ( + )a1  1 a13  1 a2 cos(   )   dt 2 2 5 cn da2   a2  a1 sin(   )  dt 2   d  (   )a2  7 a23  5 a1 cos(   ) dt  

(26)

In the following study, we consider Vdc =4V , and the parameter values used in Eq (26) are given in Table 1

Table 1. The parameter values used in Eq. (26)

parameter

cn

1

5

1

7

4V

0.1

-0.753

-0.746

-743

-807

Jo

value

Vdc

Considering da1 dT2  d dT2  da2 dT2  d dT2  0 , the frequency response equations can be derived as

[

cn2  2  (     7 a22 ) 2 ]a22  52 a12 4  

(27) 12

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  2

)a1 

1 

a13  [(   )a2 

7 

a23 ]

c 1a2 2 cn f2 ]  ( a1  1 n a22 ) 2   5 a1 2 2 5 a1 4 2

(28)

urn al P

re-

pro

of

[( 

Fig. 6 Combined frequency response curves of the electrostatic coupled resonant sensor when h2  1.01μm .( the lines present the theoretical results and the points present the numerical results; the black represents the amplitude of microbeam 1 and the red represents the amplitude of microbeam 2 )

Li et al. [40] found that the isolated branches appear in the amplitude-frequency response curve under the coupled vibration, which can lead to multiple resonance frequencies. We focus on the physical conditions and complex dynamic behaviors under multiple resonant frequencies. Firstly, the combined frequency response curves under different AC voltages are shown in Fig. 6. Considering Vac  0.76V , the amplitude-frequency response curves of microbeam 1 and microbeam 2 are similar to those of the Duffing system. With the increase of driving force, complex

Jo

saddle node bifurcation appears for Vac  0.98V . It is found that the isolated branches appear, which leads to complex jumping behaviors. Then, considering Vac  1.24V , the isolated branch disappears. Typically, there is a closed loop in the response curve of microbeam 1, as shown in Fig. 6 (c). Following, we investigate why isolated branches appear. As shown in Fig. 6 (b), there are multiple resonant frequencies in the system. We try to obtain the physical conditions of different bifurcation 13

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behaviors by analyzing the number of resonant frequencies. Through Eq. (28), the backbone curve can be decided by

a12 () 



1

[(   )a2 

7 

a23 ]

1a2     (  ) 2  5 a1 2 1

(29)

Frequency response of Eq. (28) is similar to that of Duffing oscillator. When the vibration amplitude time, substituting Eq. (29) into Eq. (28), yields

cn c f2 a1  1 n a22 )2  2 2 5 a1 42

Then, substituting Eq. (30) into Eq. (27), yields

1cn 5 2 2

[

pro

(

of

of microbeam 1 is equal to that of the backbone curve, the resonance of the system occurs. At this

c f  cn2  (     7 a22 ) 2 ]   n 4  2a1 2

(30)

(31)

Finally, we can obtain mathematical expression of resonance frequency

7 

a22 

1cn 5

(

c2 c f  n) n 2a1 2 4

re-

   

2 2

(32)

Pseudo-arc numerical continuation technique is introduced to solve Eqs. (29)- (30). We can obtain the relationship between the peak amplitudes and the driving voltage [44]. Then, substituting the

urn al P

peak amplitudes and driving voltage into Eq. (32), the resonance frequencies of the coupled resonator versus the AC voltages can be obtained, as shown in Fig. 7. When the AC voltage is below the critical value (Region A), there is only one resonance frequency in the resonator. As the driving force increases, the resonance frequency of the system increases gradually. When the AC voltage is more than the critical value (Region B), there are multiple resonance frequencies in the system. Besides, the isolated branches occur, as shown in Fig.6 (b). As the driving force continues to increase (Region C), the multi-resonance frequencies phenomenon disappears and a closed loop

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appears, as shown in Fig.6 (c).

14

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Fig. 7 Variation of the resonance frequencies of coupled resonators with different AC voltages

Then, we select three frequencies 394.7 kHz, 395.5 kHz, 395.7 kHz corresponding to a b c respectively in Fig. 7 (b) and obtain force-amplitude curves respectively, as shown in Fig. 8. The pseudo-arc numerical continuation technique is utilized to solve Eqs. (27)- (28) and the relationship between the amplitudes and the driving voltage can be obtained [44]. (1) At point a, there is no isolated branch in the amplitude-frequency response curve of the system. The force amplitude curve

of

is similar to that of the Duffing oscillator, as shown in Fig. 8 (a). (2) At point b, with the increase of driving force, the isolated branches will appear. When the AC voltage exceeds the critical value,

pro

there are three periodic solutions in the system. As the AC voltage continues to increase, the system may have five periodic solutions, as shown in Fig. 8 (b). (3) At point c, the drive frequency is close to the critical frequency. As shown in Fig. 6 (c), when the AC voltage exceeds the critical value, a closed loop appears, which can lead to very complex bifurcation behaviors of the system. Multiple bifurcation points appear in the force amplitude curves, as shown in Fig. 8 (c). To validate the above

re-

theoretical analysis results, long-time integration of Eqs. (11)- (12) is introduced to obtain some numerical solutions, compared with the analytical solution derived from the method of multiple scales. The numerical results are expressed with dot and triangle, as shown in Fig. 6 and Fig. 8. We

Jo

urn al P

find that the analytical solutions agree well with the numerical results.

15

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Fig.8 Comparison of the force-amplitude curves obtained by pseudo-arc numerical continuation technique and long-time integration method (solid line: stable; dashed line: unstable; triangle: sweeping up the frequency; dot: sweeping down the frequency; the black line represents the amplitude of microbeam 1 and the red line represents the amplitude of microbeam 2)

From the above, the phenomenon of multi-resonance frequencies is very important to the dynamic behavior of the system. Then, the physical conditions for this phenomenon are obtained. From Fig. 7, we know that the parameter space (Region B) has two critical voltages. From Eq. (32), we found

   

2 2

(



a22

c2 c f  n )  n =0 2a1 2 4

pro

1cn 5

7

of

that the AC voltage reaches the upper critical value when the square root is 0. We can obtain (33) (34)

Combining Eqs.(27), (33) and (34), the maximum driving voltage is obtained when multiple resonance frequencies phenomenon occurs, as shown in Fig. 9.

Following, we try to obtain the minimum driving voltage when multiple resonance frequencies

re-

phenomenon occurs. Working out the analytical expression is impractical, but accurate approximations can be obtained when the amplitude of microbeam 2 is small enough. We can approximately assume that when the resonance frequency of the system is equal to  2 , the

urn al P

phenomenon of multi-resonance frequency begins to occur [40]. Then we can obtain the lower critical voltage. As shown in Fig.9, the classical amplitude-frequency response curves appear in the region A. In region B and C, the isolated branches and closed loop can occur in the system, respectively. Besides, with the increase of the thicknesses of microbeam 2, the minimum critical

Jo

driving force and maximum critical driving force increase.

Fig.9 Variation of the bifurcation behavior versus h2 and Vac

5. Mass detection In section 4, the complex bifurcation behaviors of electrically coupled resonators are studied in 16

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detail. We obtain the physical conditions where the isolated branches occur. The sudden jumps in amplitude can improve the sensitivity of the system and make the detection of a very small mass possible [37]. Thus, we try to utilize the complex bifurcation jumping behaviors to realize the mass quantitative detection and threshold detection. 5.1 Quantitative detection

5.1.1 Detection Principle

of

The high sensitivity sensor can by realized by using mode localization [3]. The added mass can change the resonance peak of the system. Then, the researchers can identify the additional mass by

pro

measuring amplitude ratio of two resonant beams. However, nonlinear vibration behaviors are often ignored, which seriously affected the measurement results. Here, we present an effort to explore the exploitation of nonlinear jumping phenomenon in MEMS to realize mass sensing. First of all, the principle of mass detection is given as follow:

(1) We study variation of the amplitude-frequency responses of coupled microbeam resonators with

re-

various values of the adsorption mass, as shown in Fig.10. It is found that the adsorption mass can change the resonance peak. Then, the amplitude ratio between microbeam 2 and microbeam 1 is defined.

a2 a1

(35)

urn al P



Fig. 10 Combined frequency response curves of the electrostatic coupled resonant sensor when h2  1.01μm , Vac  1.08V ( the red lines:microbeam2; the black lines: microbeam 1 )

Jo

(2) Using the physical parameter values in the above, we can obtain the dimensionless coefficients 9 , 7 , 1 , cn , 1 ,  .

(3) The frequency response curves can be obtained by sweeping up the frequency. When the nonlinear jump occurs, we can obtain the peak amplitudes a1 , a2 (4) Substituting Eqs. (27) and (35) into Eq. (29), we can obtain the dimensionless mass   with the amplitude ratio. 17

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 

 cn2  2   52   (  2  1)  1 a12  7 a22     2 2       4 

(36)

Here, the resonance frequency is approximately equal to 1 . Thus, Eq. (36) can be rewritten as

 

 c2 2   52  2  2 2  2 2  n (   1)  1 a1  7 a2    4 1  1  1 1 

(37)

of

With Eq. (9), we can obtain the dimensionless mass   just by measuring the peak amplitude. (5) Finally, we can obtain the adsorption mass m with dimensional transformation.

(38)

pro

m= L  A1 1,12 ( L1 L) 5.1.2 Numerical experiment

The numerical experiment is introduced to prove the mass identification method. Fig. 11-12 show frequency response curves obtained by sweeping up the frequency. The blue signs correspond to the dynamic bifurcation area. When m  3  10-6 μg , the system has two bifurcation jump points, as

re-

shown in Fig. 11. With the increase of adsorption mass, only one bifurcation jump point appears in the system, as shown in Fig. 12. The added mass can affect the vibration amplitude of the bifurcation point. The bifurcation amplitudes under different mass perturbation are obtained by numerical experiment in Fig. 13. Using the above mass detection method, we obtain four kinds of

urn al P

parameter identification results, as shown in Table 2. The results show that the mass detection method presented in this paper can accurately identify the adsorption mass. It should be noted that

Jo

the accuracy of identification increases with the increase of additional mass.

Fig.11 Combined frequency response and swept harmonic responses of the resonators for m  3  10-6 μg , Vac  1.24V , h2  1.01μm and L1  64μm

18

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urn al P

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Fig.12 Combined frequency response and swept harmonic responses of the resonators for m  1.5  10-5 μg , Vac  1.24V , h2  1.01μm and L1  64μm

Fig. 13 The dimensionless amplitudes of the microbeam 1 and microbeam 2 as a function of the mass perturbation Table 2 Four groups of mass detection results with the quantitative detection method presented in this paper

Number

The true mass

Dimensionless

Identification

amplitude a1

amplitude a2

results

Error

m

(10-5 μg)

1

1

0.124

0.0773

1.09

9%

2

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m (10-5 μg)

Dimensionless

1.5

0.127

0.0767

1.526

1.7%

2

0.130

0.0742

2.021

1.05%

2.5

0.133

0.0729

2.516

0.64%

3 4

To study the influence of nonlinear factors on the sensor, we give a mass detection formula without 19

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considering nonlinear behavior

linear 

 cn2 2   52  (  2  1)     2 2     4 

(39)

Then, the nonlinear error functions is defined

linear    100% 

(40)

of

e

Fig.14 shows the variation of nonlinear detection error under different mass. It is found that the nonlinear behavior is crucial to the accuracy of mass detection. When the adsorption mass is very

pro

small, the nonlinear detection error reaches 20%. As the adsorption mass increases, the value of the error function decreases. Thus, when detecting small mass adsorption, nonlinear vibration behavior

urn al P

re-

cannot be ignored.

Fig.14 Variation of nonlinear detection error under different mass.

5.2 Threshold detection

5.2.1 Detection Principle

In this section, we try to utilize the jump phenomena of the isolated branches to realize mass threshold detection. Younis et al. [14] presented an effort to utilize dynamic instabilities and

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bifurcations in MEMS to realize the mass sensing and detection. The added mass can lead to dynamic jump phenomena near the bifurcation point, which leads to an amplified response of the excited structure making the sensor more sensitive and its signal easier to be measured. Similarly, we use two microbeams to realize the mass threshold detection. In this way, the adsorption mass of microbeam 1 can be obtained by detecting the dynamic behavior of microbeam 2, which can reduce the impact of ambient noise on the system. First of all, the principle of mass threshold detection is given as follow: 20

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(1) We study variation of the resonance frequencies of electrically coupled resonators with various values of the AC voltages and adsorption mass, as shown in Fig.15. We find that the adsorption mass can reduce the resonance frequency and change the parameter space where the isolated branches occur. If the system is operating in a nonlinear vibration range, amplitude jump phenomenon may occur when the adsorption mass is added. (2) Then, the drive voltage and drive frequency of the system is critical to realize mass detection. resonance frequency in Eq. (32).

7 

a22 

1cn 5 2 2

(

c2 c f  n) n 2a1 2 4

pro

   

of

The design drive frequency under different mass threshold can be obtained by choosing a smaller

(41)

where a1 and a2 can be affected by the adsorption mass and they can be obtained by Eq (29) and (30). For example, we select the operating voltage of the system Vac  1.2V in Fig. 15. Point a, b and c represent the resonance frequencies of the system under different adsorption mass. When no

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adsorption mass is added, there is only one resonant frequency in the system under Vac  1.2V .

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When m=1.5×10-5g and m=3×10-5g, there are multiple resonant frequencies in the system.

Fig. 15 Variation of the resonance frequencies of coupled microbeam resonators with various values of the AC voltages when L1  64μm

(3) Following, we choose the appropriate operating frequency by using Eq. (41). To identify the

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presence of the added mass, the microbeam is forced at an operating frequency close to the bifurcation frequency. Here, the driving frequency corresponding to point c is selected. (4) In practice, as the adsorption mass increases, the isolated branches appear. When the adsorption mass exceeds the critical value m=3×10-5g, the sudden presence of the added mass induces a dynamic jump from point A1 in Fig.16 (a) (without added mass) to point A2 in Fig.16 (b) (with added mass). 21

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Fig.16 Combined frequency response curves of the resonators when L1  64μm (the black line represents the amplitude of microbeam 1 and the red line represents the amplitude of microbeam 2)

5.2.2 Numerical experiment

Fig.17 shows design drive frequency under different detection threshold obtained by Eq. (41). The

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numerical experiment is introduced to prove the above mass threshold detection method. Fig. 18 and 19 show response curves obtained by sweeping up the added mass. The blue signs correspond to the dynamic bifurcation area. As the adsorption mass increases, the vibration amplitude of microbeam 1 increases gradually and the vibration amplitude of microbeam 2 has no obvious

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change. When the adsorption mass exceeds the critical value, dynamic bifurcation occurs and the amplitude suddenly decreases. Here, we obtain three kinds of mass detection results, as shown in Table 3. The results show that the mass threshold detection method presented in this paper can identify the added mass. However, the result of numerical experiment is smaller than the real mass. When the adsorption mass is near the critical value, the attractive region of steady state vibration becomes very small. Small numerical perturbations can lead to dynamic bifurcation in advance, which make the result of numerical experiment is smaller than that of theoretical prediction. In this section, bifurcation jumping behavior of coupled systems is introduced to realize the mass detection. The sudden amplitude changes of the bifurcation point can improve the sensitivity of the sensor. Besides, the bifurcation-based detection method presented in this article can be extended to the detection of environmental signals and implementation of switch. In a word, it is found that the

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complex bifurcation behavior under coupled vibration has obvious advantages to detection by exploiting the bifurcation jumping behavior, which provides a theoretical guarantee for the further development of nonlinear mass sensors.

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Fig.17 Design drive frequency under different mass threshold

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Fig.18 Variation of the amplitudes of coupled microbeam resonators with the increase of added mass for h2  1.01μm , L1  64μm Vdc  4V and Vac  1.2V

Fig.19 Variation of the displacements of coupled microbeam resonators with the increase of added mass for h2  1.01μm , L1  64μm Vdc  4V and Vac  1.2V (a) the displacement of microbeam 1; (b) the displacement of microbeam 2 Table 3 Three groups of mass detection results with the threshold detection method presented in this paper

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Number

The true mass m (10-5 μg)

Driving frequency

AC voltage (V)

(kHz)

Identification results

Error

m

(10-5 μg)

3.75

396.3

1.2

3.585

4.4%

2

3

396.6

1.2

2.865

4.5%

3

2.25

397

1.2

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1

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6. Conclusion

5.3%

In this paper, the Hamilton’s principle and Galerkin discretization are applied to obtain two degrees of freedom equations of the coupled resonant structure. The microbeams with slightly different thicknesses are chosen to realize two close but different frequencies of the system. Then, the method of multiple scales is used to investigate the response of the MEMS resonators with small

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amplitude vibration around equilibrium position. Through bifurcation analysis, the closed loop and isolated branches of amplitude-frequency response curves are found. Then, the physical conditions of isolated branches and closed loop of the two-degree-of-freedom system are deduced by using nonlinear dynamics theory, which provides a reference for the design of the coupling resonator.

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Besides, we present the new mass quantitative detection method and threshold detection method with the complex bifurcation behavior in coupled vibration system. We utilize the amplitude ratio of bifurcation jump points to realize the mass quantitative detection and analyze the influence of nonlinear behavior on mass quantitative detection. The results show that the nonlinear behavior cannot be ignored under the condition of small mass detection. Meanwhile, with the increase of adsorption mass, the accuracy of detection is improved. Besides, the bifurcation jump is used to realize the mass threshold detection. By designing different driving frequency, different mass threshold detection can be realized. The coupled resonant structure has obvious advantages to parameter identification, which may be useful to improve sensor. The framework presented here overcomes many problems of accurately predicting complex dynamics in MEMS. It should be emphasized that all the theoretical results in this paper are compared with numerical results, which

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guarantees the accuracy of our whole investigations.

Acknowledgments

The work was supported by the National Natural Science Foundation of China (Grant No. 11902182 and 11802173), National Science Foundation for Distinguished Young Scholars (11625208), Natural Science Foundation of Shandong Province (ZR2019BA001) and the China Postdoctoral Science Foundation（2019M651485） 24

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Conflicts of Interest: The authors declare no conflict of interest

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Table 1. The parameter values used in Eq. (26) Vdc

cn

1

5

1

7

value

4V

0.1

-0.753

-0.746

-743

-807

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parameter

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Table 2 Four groups of mass detection results with the quantitative detection method presented in this paper

true Dimensionless Dimensionless

mass

amplitude a1

amplitude a2

Identification Error results

m

(10-5 μg)

m (10-5 μg)

of

Number The

1

0.124

0.0773

1.09

9%

2

1.5

0.127

0.0767

1.526

1.7%

3

2

0.130

0.0742

2.021

1.05%

4

2.5

0.133

0.0729

2.516

0.64%

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Table 3 Three groups of mass detection results with the threshold detection method presented in this paper

The

AC voltage

true Driving

mass m (10-5 μg)

frequency (kHz)

(V)

Identification Error results

m

(10-5 μg)

of

Number

3.75

396.3

1.2

3.585

4.4%

2

3

396.6

1.2

2.865

4.5%

3

2.25

397

1.2

2.13

5.3%

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Highlights

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1. We derive the physical conditions for different bifurcation behaviors. The closed loop and isolated branches of amplitude-frequency response curves are found. 2. The amplitude ratio of bifurcation jump point is utilized to realize the mass quantitative detection. 3. The bifurcation jump is used to realize the mass threshold detection. 4. Numerical experiments are introduced to prove the accuracy of the parameter identification method

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Conflicts of Interest:

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The authors declare no conflict of interest