Multi-field coupled chaotic vibration for a micro resonant pressure sensor

Accepted Manuscript

Multi-field Coupled Chaotic Vibration for a Micro Resonant Pressure Sensor Xiaorui Fu , Lizhong Xu PII: DOI: Reference:

S0307-904X(19)30179-9 https://doi.org/10.1016/j.apm.2019.03.035 APM 12736

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

16 December 2018 4 March 2019 26 March 2019

Please cite this article as: Xiaorui Fu , Lizhong Xu , Multi-field Coupled Chaotic Vibration for a Micro Resonant Pressure Sensor, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.03.035

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ACCEPTED MANUSCRIPT Highlights Multi-field coupled dynamic model of a micro resonant pressure sensor is proposed. The multi-field coupled chaotic vibrations of the sensor are investigated. Effects of the multi-physics field parameters on the vibration states of the sensor are determined.

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Multi-field Coupled Chaotic Vibration for a Micro Resonant Pressure Sensor Xiaorui Fu Lizhong Xu* Mechanical engineering institute, Yanshan University, Qinhuangdao, China

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Abstract A multi-field coupled dynamic model for a micro resonant pressure sensor is proposed in this paper which considers molecular force, electric field force, elastic force, and air damping force. Using these equations, the nonlinear dynamical performance of the sensor is investigated. The results show that the initial clearance between the resonator and baseplate has an important effect on the vibration states of the sensor for a given voltage. For small clearance sensors, the amplitude and range of displacement oscillating becomes large when molecular force is considered. The operating pressure can influence the vibration states of the sensor, and small dynamic viscosity easily induces chaotic vibrations. Thus, the related multi-field parameters must be selected carefully to avoid harmful vibration states of the sensor. Keywords: Micro-resonant pressure sensor; multi-field coupled; chaotic vibration; resonator

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1. Introduction

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Sensors are detection devices that can convert physical properties into electric signals [1]. With the development of silicon micromachining technology, sensors are becoming increasingly important and widely used in fields such as robotics, smart devices, and medical devices [2-3]. Silicon micro resonance sensors have been widely studied due to possessing benefits such as non-distorted output frequency signals, easy measurement, and suitability for long-distance transmission [4-5]. The vibration theory of the resonator was developed in the 18th century from the theory of string vibration [6]. Katsikadelis [7] developed a boundary element method (BEM) based method to analyze the dynamic response of orthotropic membranes. Fan [8] investigated the nonlinear vibration characteristics of the resonant sensor under one and two frequency excitations, in which the nonlinear characteristics of the sensor were analyzed, and a method to improve the performance of the micro resonant silicon sensor was developed. Reutskiy [9] presented a new numerical method for analyzing the free vibrations of the arbitrary shaped membranes in a method based on the mathematical modeling of membrane response to excitation signals. Li investigated the nonlinear dynamics of a resonant pressure sensor under heat excitation. In this research, Galerkin and perturbation methods were used to determine the approximate solutions of nonlinear dynamics equations [10]. Tajaddodianfar [11] proposed a Homotopy Analysis Method (HAM) for derivation of analytical solutions for the frequency response of the resonators, and investigated the nonlinear dynamics of electrostatically actuated micro-resonators. Due to nonlinearities in the micro resonance sensor, chaotic vibration may occur, and study into this phenomenon is required. Tiago [12] analyzed the chaotic vibration of a fixed beam resonator by the one degree of freedom (1-DOF) model, considering geometric nonlinearity and the Casimir force. Miandoab [13] proposed a novel method for predicting the chaos of the micro resonators, developing a technique that can be used in design of the micro-resonators. Shi [14] designed a harmonic oscillator, and developed a new detection method to avoid chaotic noise interference of the sensors. Alemansour [15] investigated the size effect on chaotic vibration of a micro resonator under electrostatic excitation by Galerkin's decomposition method and Melnikov's method. Pishkenari [16] studied the chaotic vibrations of an electrostatically actuated arch 2 * corresponding author: [email protected]

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resonator by the Melnikov’s method. Bassinello [17] analyzed the influence of the system parameters on the chaotic vibrations of the vibration system using phase diagram, bifurcation diagram, and the 0-1 test, and Luo [18] studied the fractional-order chaotic vibrations of the micro resonators. In a micro resonant pressure sensor, multi-physics fields occur simultaneously and include elastic force, electric field force (electrostatic excitation), air damping force, and molecular force. These fields can induce complicated dynamics performance of the sensor which will disturb its normal operation and under some conditions, the chaotic vibrations are more harmful for the measure sensitivity of the sensor. It is thus important to predict the chaotic vibration of micro resonant pressure sensors and determine corresponding parameter regions. In some studies about dynamics of the MEMS elements and system, analysis and control of the chaos have been done [19-21]. However, in these studies, the lumped parameter models were used and less nonlinear factors are taken into account. In this paper, a multi-field coupled dynamic continuous model of a micro resonant pressure sensor is proposed, considering the influence of molecular force, electric field force, elastic force, and air damping force, which is barely explored in the literature. Using these equations, the nonlinear dynamical performance of the sensor is investigated. The results show that the initial clearance between the resonator and baseplate has an important effect on the vibration states of the sensor for a given voltage. For small clearance sensors, amplitudes and range of the chaotic vibrations becomes large when molecular force is considered. The operating pressure could influence vibration states of the sensor as well. Small dynamic viscosity can easily induce chaotic vibrations in the sensor. These results can be used to design the micro resonant pressure sensor to avoid chaotic vibrations.

2. Multi-field Coupled Nonlinear Dynamics Equations

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The structure model of a micro resonant pressure sensor and the multi-field coupled nonlinear dynamics model of its resonator are provided in Fig. 1. The molecular force, electric field force, and air damping force are applied to the micro resonant film. Here, a and b denote length and width of the resonant film, h is thickness, Δw is the vibration displacement in the z-axis direction, and v is the clearance between the resonator and baseplate. According to the D'Alembert principle [22], the nonlinear-dynamics equation and the control equation of an orthotropic micro resonant film are [23]: Resonator

Conductive layer

Top

a

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 0x

~

Bottom

Membrane

x

C

 0x

b/2

a/2

U

Fixed plate

0

b z h

+

B A

UU0 ·

y

Diaphragm Measured Pressure

Δ q(x,y,t)

v

TO8 socket

(a) Structure model

(b) dynamics model

Fig. 1: Structure and dynamics models of a micro resonant pressure sensor

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 2 w w   2   2 w   2   2 w      h     q  x0   y0 h t 2 t  y 2  x 2  x 2  y 2 3

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(1)

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1  4 1  4   2 w   2 w  2 w     E1 y 4 E2 x 4  xy  x 2 y 2

(2)

where 𝜌 is the density of the micro resonator, 𝑈0 is the static voltage between the resonator and baseplate, 𝜑 denotes the stress function, and 𝜎x0 and 𝜎𝑦0 are the initial tension stress in 𝑥 direction and 𝑦 direction of the resonator. The E1 and E2 are Young's moduli in x direction and y direction, and  denotes the damping coefficient between film and fixed plate, which is viscoelastic damping due to squeeze film damping and can be calculated as [24]

 ab3

 v  w0 

3

μ =1.86×10−5N ⋅ S ⋅m−2 .

where μ is the air dynamic viscosity,

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

(3)

q is the force per unit area on the resonator, it includes three components

q=q1  q2  q3

(4)

q1 

60  v  w0 

w  5

 2 c 5

12v w0

w2 

 2 c 6

4v w0

w3 

(5)

8  1.055 1034 J  S ; c is light speed, c  2.998 10 m s .

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where

2 c

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From the Lifshitz formula, Van der Waals' force per unit area on the micro resonator can be obtained [25]

Dynamic electrostatic force per unit area on the resonator caused by static voltage is [26]

U 0 2 0

w  3

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q2 

 v  w0 

3U 0 2 0 6U 0 2 0 2  w  w3  3 4 v w0 v w0

(6)

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where 𝜀0 is the dielectric constant of air，  0  8.85  1012 c2  N -1  m-2 When excitation signal V  E0 cos  f t is applied between the resonator and fixed plate,

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the electric force per unit area is

q3 

U 0 E0 0 cos  f t

v  w 

2

(7)

0

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Substituting Eqs. (5), (6) and (7) into Eq (4), yields

 U 2  3U 0 2 0  2 c  2 c  2 q =  0 0 3   w     3  w  5 5 v w 12 v w   v  w0  60  v  w0   0 0  U E  cos  f t  6U 0 2 0  2 c   6  w3   0 0 0  4 2 4v w0   v  w0   v w0

Substituting Eq. (8) into Eq (1), one obtains

4 * corresponding author: [email protected]

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 2 w w   2   2 w   2   2 w      h      x0   y0 h t 2 t  y 2  x 2  x 2  y 2  U 2  3U 0 2 0  2 c   6U 0 2 0  2 c  2 c  2  w     w   6  w3  0 03   3   4 5 5 v w 12 v w v w 4v w0    v  w0  60  v  w0   0 0 0   U E  cos  f t  0 0 0 2  v  w0 

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(9)

The solutions of Eq. (1) and Eq. (2) can be given as [27]

w( x, y, t )  W ( x, y )T (t )

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(10)

 ( x, y, t )   ( x, y)T (t )   m x n y 2

W ( x, y )   sin m 1 n 1

a

cos

Substituting Eq. (11) and Eq. (12) into Eq. (2) yields

(11)

(12)

b

1  4 1  4 m 2 n 2 4  2m x 2n y    cos  cos  4 4 2 2  E1 y E2 x 2a b  a b 

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Let

(13)

2m x 2n y   cos a b into Eq. (13),  and  can be given as

  x, y    cos

Substituting

  x, y 



(14)

E2 n 2 a 2 E1m2b2 ，   32m2b 2 32n2 a 2

(15)

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Substituting Eq. (15) and Eq. (14) into Eq. (11), yields

 E2 n2 a 2 2m x E1m2b2 2n y  2 cos  cos  T (t ) 2 2 2 2 a 32n a b   32m b

(16)

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 ( x, y, t )  

According to Galerkin’s method [28], it is obtained

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  2 w w   2   2 w   2   2 w  m x n y  h      h    h sin cos dxdy   0x  0y 2  2 2  2    t 2 t  y  x  x  y  a b

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  U 2   3U 0 2 0  2 c  2 c  2  0 0 3    w     w   3  5 5    v  w0  60  v  w0     v w0 12v w0  m x n y cos dxdy  sin    2 a b U E  cos  t  6U 0  0  2 c   0 0 0 f 3 2   v 4 w  4v 6 w  w     v  w0  0 0    (17)

Let

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 2 m x 2 n y  S111    h sin a cos b dxdy   S   sin 2 m x cos 2 n y dxdy  222  a b  2 2  S     W    W  h sin m x cos n y dxdy  0y  333   0 x x 2 y 2  a b     2  2W  2  2W  m x n y  2 h sin cos dxdy  S 444    2 2 2   y  x  x  y a b      U 2   2 c  2 m x n y sin cos 2 dxdy  S555    0 0 3  5 a b   v  w0  60  v  w0      3U 0 2 0  2 c   3 m x 3 n y S   666   v3 w  12v 5 w  sin a cos b dxdy 0 0    2 2   6U   c n y 4 m x cos 4 dxdy  S777    40 0  6   sin 4v w0  a b   v w0  U E m x n y  S888    0 0 0 2 sin cos dxdy a b   v  w0  Thus, Eq. (17) can be given as

S111T   S222T   S333T  S444T 3  S555T  S666T 2  S777T 3  S888 cos  f t

(18)

(19)

Equation (19) can be transformed into state-space notation, given by

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T  T 2  1 S  S555 S S  S 444 3 S888 S 222  T2  333 T1  666 T 21  777 T 1 cos T3  T2   S S S S S 111 111 111 111 111  T   f  3

(20)

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3. Results and Discussions

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From Eq. (20), the function T(t) can be obtained by by the Runge-Kutta numerical method. w( x, y, t )  W ( x, y )T (t ) W ( x, y) Substituting T(t) and into (here, 10

10

W ( x, y )   sin m 1 n 1

m x n y cos and m and n are taken to be 1-10, respectively), the dynamic a b

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displacements of the resonator can be given. In the numerical method, as irregular responses due to transient often occur on the beginning of the calculation, the first 300 periods are removed and the last 100 periods are shown. The parameters of the example sensor are provided in Table 1. Table 1. Parameters of the example sensor a

b

h.

v

0

(mm)

(mm)

(μm)

(μm)

(c N m )

2

1

5

2

2

8.85  10

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-1

-2

12

E1

E2

ρ

(GPa)

(GPa)

(kg/m3)

1.4

0.9

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ACCEPTED MANUSCRIPT In order to study the effect of initial clearance on the dynamics performance of the sensor, the global bifurcation and largest Lyapunov exponent spectrum in the range of the initial clearance v ∈[4μm, 10μm] are calculated. These are shown in Fig. 2(a) and Fig. 2(b) (here, U0=0.1v, E0=1v,

 x 0 =4.5 106 N / m and ωf=31000 rad/s).

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The local bifurcation and the largest Lyapunov exponent (LLE) spectrum in the range of film initial clearance v∈[8.2 μm, 8.3 μm] is illustrated in Fig. 3. To precisely describe the dynamic behaviors, Fig. 4 shows the time-domain waveform, phase diagram and Poincaré map, and power spectrum when the clearance is equal to 8.222 μm, 8.240 μm, 8.260 μm, and 8.280 μm. Figures 2-4 show the following:

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Fig. 2 Global bifurcation(a) and largest Lyapunov exponent spectrum(b) for v∈[4μm, 10μm]

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(1) The clearance between the resonator and the substrate has a significant effect on the dynamic

characteristics of the sensor. According to the bifurcation diagram and the maximum Lyapunov exponent, the vibration of the resonator changes from periodic vibration to chaotic vibration with the increase of clearance, and the chaotic vibration is transformed into periodic vibration. This conversion occurs many times. For small clearance（below 5.67 μm)，the amplitude of the resonator is determined, and the maximum Lyapunov exponent is negative. The vibration of the resonator is fixed period. When the initial clearance v=5.67 μm, the largest Lyapunov exponent becomes positive, and there are multiple unstable vibration amplitudes at the corresponding points on the bifurcation diagram. This indicates that the resonator vibration is no longer periodic, and has transformed into the chaotic vibration state. When the initial clearance v=5.67-6.3 μm, the bifurcation diagram is in the multi-amplitude 7

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oscillation region, the maximum Lyapunov exponent appears alternately positive and negative, and the negative region is obviously more than the positive region. From the above characteristics, it can be determined that the periodic vibration, quasi-periodic vibration, and chaotic vibration occur alternately in the range of clearance variation, and the region of chaotic vibration is relatively small. When the initial clearance v=6.3-7.5 μm, the maximum Lyapunov exponent is negative and the bifurcation diagram is of two lines, indicating that the vibration of the resonator is biperiodic in the clearance range. When the initial clearance v is above 7.5 μm, the multi-amplitude oscillation region appears again on the bifurcation diagram, and the amplitude oscillation range increases noticeably when v is more than 8 μm. The maximum Lyapunov exponent image shows that the positive and negative values appear alternately, and the positive region is much larger than the negative region. The main results are as follows: in the range of clearance variation, the periodic vibration, quasi-periodic vibration, and chaotic vibration occur alternately, and the region of chaotic vibration increases obviously. The local bifurcation diagram and the maximum Lyapunov exponent diagram presented in Fig. 3 show that the resonator moves into the chaotic state by period doubling bifurcation. In Fig. 4(a), distributed point sets with no ring shape appear on the Poincaré map, irregular reciprocating motions appear on the phase map, and the power spectra becomes continuous (here, the corresponding largest Lyapunov exponent value is 0.0893, see Fig. 3(b)). These findings further illustrate that chaotic vibrations occur at v=8.222 μm. In Fig. 4(b), there are eight closed curves, eight isolated points occur on the corresponding Poincaré map, and the power spectrum is discrete. The evidence suggests that the period-8 vibration occurs at v=8.245 μm (here, the corresponding largest Lyapunov exponent value is -0.0157, see Fig. 3(b)). These results further demonstrate that the period-8 vibrations occur at v=8.24 μm. In Fig. 4(c), in which the corresponding largest Lyapunov exponent value is negative (-0.005), the periodical vibrations could be found in the time-domain waveform. In the phase diagram, there are four closed curves, four isolated points occur on the corresponding Poincaré map, and the power spectrum is discrete. The evidence suggests that the period-4 vibration occurs at v=8.260 μm. In a same manner, as indicated in Fig. 4(d), the period-2 vibration occurs at v=8.280 μm (here, the corresponding largest Lyapunov exponent value is -0.006) . Therefore, it is proved that the resonator moves into the chaotic state by period doubling bifurcation (period-2 period-4 period-8 … period-2n Chaos). These results demonstrate the effects of electrostatic nonlinearity on system stiffness of the resonant sensor. The electrostatic force is equivalent to a soft spring that causes a decrease in resonator stiffness. As the clearance between the resonator and baseplate is reduced, the electrostatic force is increased which correspondingly amplifies electrostatic nonlinearity. This finding illustrates that stiffness of the resonator decreases more significantly when the clearance between the resonator and baseplate is reduced. To summarize, the initial clearance between the resonator and the baseplate has an important effect on the nonlinear vibration of the micro resonant pressure sensor system. To avoid unreliable dynamic performance of the resonant sensor, the initial clearance must be properly selected.

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x 10

-9

0.04

2

0 -1

0.03

2 Spectrum

dw/dt/(m/s)

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w/m

-5

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0.01

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dw/dt/(m/s)

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0.2 0.15

2 dw/dt/(m/s)

0.065

w/m

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0.01

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Spectrum

w/m

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0.015

Spectrum

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0.065

0.07 t/s

0.075

-10

-5

0

w/m

5 -10 x 10

0 0

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(d)

Fig. 4 Time-domain waveform, phase diagram and Poincaré map, power spectrum for different initial clearance:

(a) 8.222μm; (b) 8.240μm; (c) 8.260μm; (d) 8.280μm

Figure 5 shows the global bifurcation and the largest Lyapunov exponent spectrum for the initial tension stress  x 0 in resonator varying from 0 to 10╳106N/ m2. Figure 6 gives the local bifurcation and the largest Lyapunov exponent(LLE) spectrum in the range of initial tension stress  x 0 ∈[2.0╳106N/ m2, 2.4╳106N/ m2]. Figure 7 shows the time-domain waveform, phase diagram and Poincaré map, and power spectrum when the initial tension stress is equal to 2.1╳106N/ m2, 9 * corresponding author: [email protected]

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2.2╳106N/ m2, 2.3╳106N/ m2 and 2.4╳106N/ m2 . From these figures, the following is known: (1) As the initial tension stress increases, the positive and negative largest Lyapunov exponents appear alternately, and the chaotic vibration and periodic vibration window appear alternately in the bifurcation diagram as well. When the stress value is less than 2Mpa, the resonator is basically in chaotic vibration state. When the stress value is above 2Mpa, the largest Lyapunov exponent becomes negative, the bifurcation diagram corresponds to the fixed period vibration. The periodic vibration window ends when the stress is equal to 2.5Mpa, and the resonator enters the chaotic vibration state again. When the stress varies from 4.3Mpa to 6.1Mpa, a large range of periodic vibration window appears again. When the stress is above 6.1Mpa, the resonator is mainly in the state of chaotic vibration, and there are many narrow periodic vibration windows in the middle. The local bifurcation diagram and the largest Lyapunov exponent diagram presented in Fig. 6 show that the chaotic vibration of resonator due to stress change is also by period-doubling bifurcation (PDB). The time domain waveform, phase diagram and Poincare map for the four special stress values presented in Fig. 7 show that when the stress is equal to 2.1╳106N/ m2, there are not periodic vibrations in the time domain waveform; there is a singular attractor in the phase diagram; there is a non circular distribution lattice in the Poincare map, and a continuous power spectrum. These characteristics further prove that the resonator is in a chaotic state under this stress. At σx0=2.2╳106N/ m2, there are eight closed curves in the phase diagram, eight isolated points occur on the corresponding Poincaré map, and the power spectrum is discrete. The evidence suggests that the period-8 vibration occurs at σx0=2.2╳106N/ m2 (here, the corresponding largest Lyapunov exponent value is negative, see Fig. 6(b)). At σx0=2.3╳106N/ m2, there are four closed curves in the phase diagram, four isolated points occur on the corresponding Poincaré map, and the power spectrum is discrete. The evidence suggests that the period-4 vibration occurs at σx0=2.3 6 2 ╳10 N/ m (here, the corresponding largest Lyapunov exponent value is negative, see Fig. 6(b)). At σx0=2.4╳106N/ m2, there are two closed curves in the phase diagram, two isolated points occur on the corresponding Poincaré map, and the power spectrum is discrete. The evidence suggests that the period-2 vibration occurs at σx0=2.4╳106N/ m2 (here, the corresponding largest Lyapunov exponent value is negative, see Fig. 6(b)). The results proved that the resonator moves into the chaotic state by period doubling bifurcation when the initial tension stress is changed. (2) The initial tension stress  x 0 in resonator has significant effects on the dynamics performance of the resonator. The initial tension stress in resonator reflects pressure amplitudes measured by the resonant pressure sensor (see Appendix A). Above-mentioned results show that different operating pressure could also influence vibration states of the sensor as the initial tension stress can change stiffness of the resonator. When the initial tension stress is increased, the stiffness of the resonator is increased as well. It can change frequency zone for chaotic vibration and cause complicated vibration state harmful to the sensor operation. Hence, the resonator parameters must be taken properly for a given pressure range measured in order to avoid these harmful vibration states. 0.3

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 /N/m x

10 * corresponding author: [email protected]

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(a) (b) Fig.5 Global bifurcation (a) and largest Lyapunov exponent spectrum(b) in the range of initial tension stress

 x 0 ∈[0, 10╳106 N/m2] 0.25 0.2 0.15

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(a) (b) Fig.6 Local bifurcation (a) and Largest Lyapunov exponent spectrum(b) in the range of initial tension stress

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(d) Fig.7 Time-domain waveform, phase diagram and Poincaré map, Power spectrum for different initial tension stress: (a) 2.1╳106N/ m2; (b) 2.2╳106N/ m2; (c) 2.3╳106N/ m2 (d) 2.4╳106N/ m2

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As everyone knows, sizes of the resonator have important effects on the vibration states for the resonant sensors. Length, width and height of the resonator have effects on the vibration states for the resonant sensors. To reduce word number, only the effects of the resonator width on the vibration states were given here. Figure 8 shows the global bifurcation and the largest Lyapunov exponent spectrum for resonator width b varying from 1 mm to 1.8 mm. The local bifurcation and the largest Lyapunov exponent(LLE) spectrum for resonator width b varying from 1.43 mm to 1.46 mm are shown in Fig. 9. Figure 10 shows the time-domain waveform, phase diagram and Poincaré map, and power spectrum when the resonator width b is equal to 1.435mm, 1.445mm, 1.451mm and 1.455mm. From these figures, the following is known: (1) As the width of the resonator is small，the largest Lyapunov exponent is negative，and the vibration amplitude of resonator in bifurcation diagram is determined. This shows that the resonator is in periodic vibration states. When the resonator width b=1.022mm, the largest Lyapunov exponent λ>0, and there are a lot of unstable vibration amplitudes in the corresponding points on the bifurcation diagram, indicating that the resonator vibration is no longer periodic and has entered the chaotic vibration state. (2) For resonator width b varying from 1.022 mm to 1.664 mm，the periodic vibration and chaotic vibration are alternately oscillated multi-times. Here, chaotic vibration is the main form of vibration, and the periodic vibration occurs under some width parameters. For example, when b=1.088mm and 1.439mm, the largest Lyapunov exponent becomes negative, and the resonator moves into the periodic vibration state. (3) When the width of the resonator is above 1.664mm，the largest Lyapunov Exponent becomes negative，and one period vibration occurs in the bifurcation diagram. This shows that the large width of resonator is very beneficial to avoid chaotic vibration. (4) The local bifurcation diagram and the largest Lyapunov exponent diagram presented in Fig. 9 show that the chaotic vibration of resonator due to the change of resonator width is caused by period-doubling bifurcation (PDB) as well. Figure 10 gives the time-domain waveform, phase diagram and Poincaré map and power spectrum of resonator motion for several typical width of the resonator. Figure 10 shows: At b=1.435mm, distributed point sets with no ring shape appear on the Poincaré map, irregular reciprocating motions appear on the phase map, and the power spectra becomes continuous (here, the corresponding largest Lyapunov exponent value is 0.045, see Fig. 9b). These findings further illustrate that chaotic vibrations occur at b=1.435mm. 12 * corresponding author: [email protected]

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In Fig. 10b ( b=1.445mm), there are nine closed curves, nine isolated points occur on the corresponding Poincaré map, and the power spectrum is discrete. The evidence suggests that the period-9 vibration occurs at b=1.445mm (here, the corresponding largest Lyapunov exponent value is -0.0136, see Fig. 9b). These results further demonstrate that the period-9 vibrations occur at b=1.445mm. In Fig. 10c ( b=1.451mm), there are eighteen closed curves, eighteen isolated points occur on the corresponding Poincaré map, and the power spectrum is discrete, in which the corresponding largest Lyapunov exponent value is negative, see Fig. 9b). These results further demonstrate that the period-18 vibrations occur at b=1.451mm. In Figure 10d (b=1.455mm), distributed point sets with no ring shape appear on the Poincaré map, irregular reciprocating motions appear on the phase map, and the power spectra becomes continuous (here, the corresponding largest Lyapunov exponent value is 0.048, see Fig. 9(b)). These findings further illustrate that chaotic vibrations occur again at b=1.455mm. Therefore, it is proved that the resonator moves into the chaotic state by period doubling bifurcation (period-20x9 period-21x9 … period-2nx9 Chaos). In a word, the sizes of the resonator have important effects on the vibration states of the micro resonant pressure sensor. To avoid harmful vibration states, the sizes of the resonator are effective control parameters. The sizes of the resonator have influence on both stiffness and mass of the resonator which can be used to tune vibration frequency of the sensor.

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Fig.8 Global bifurcation(a) and largest Lyapunov exponent spectrum(b) for b∈[1.0 mm, 1.8 mm]

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Fig.9 Local bifurcation(a) and largest Lyapunov exponent spectrum(b) for b∈[1.43 mm, 1.46 mm]

13 * corresponding author: [email protected]

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Fig.10 Time-domain waveform, phase diagram and Poincaré map, Power spectrum for different width: (a) 1.435mm; (b) 1.445mm; (c) 1.451mm; (d) 1.455mm

Figure 11 shows the global bifurcation and the largest Lyapunov exponent spectrum for different dynamic viscosity of gases (other diagrams are not listed due to space constraints). As can be seen from the graph: When the dynamic viscosity is less than 3.1×10−5N⋅S⋅m−2, the chaotic vibration and periodic vibration occur alternately. When the dynamic viscosity μ is above 3.1×10−5N⋅S⋅m−2, the 14 * corresponding author: [email protected]

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Lyapunov exponent becomes negative and the resonator is in periodic vibration state. Lyapunov exponent is -0.0085 and two isolated points occur on the Poincaré map for μ=3.1×10−5N⋅S⋅m−2. It shows that the period-2 vibration occurs when the dynamic viscosity μ is above 3.1×10−5N⋅S⋅m−2. The results show that small dynamic viscosity easily causes the chaotic vibrations in the sensor. As the micro resonant pressure sensor operates at vacuum (this could increase sensitivity of the sensor), the dynamic viscosity in this environment is quite small. It quite easily causes the chaotic vibrations in the sensor. Hence, the related parameters must be selected carefully to avoid harmful vibration states of the resonant sensor. 0.08

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(a) (b) (c) Fig.11 Global bifurcation(a), largest Lyapunov exponent spectrum(b) in the range of dynamic viscosity μ∈ [1×10−5N⋅S⋅m−2 , 5×10−5N⋅S⋅m−2] and Poincaré map(c)(μ=3.1×10−5N⋅S⋅m−2)

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The effects of molecular force on the nonlinear vibration of the sensor are studied (here, v ˂2μm, 𝑈0=0.01v). Figure 12 shows the largest Lyapunov exponent spectrum for different clearances with and without molecular force. It shows: In the range of the clearance considered, the positive and negative maximum Lyapunov exponents of the resonator vibrations occur alternately with the increase of the clearance, which indicates that the vibration state of resonator appears alternately between periodic vibration and chaotic vibration. When the clearance is small, the clearance range with the negative maximum Lyapunov exponent is larger. With the increase of the clearance, the range of the clearance whose maximum Lyapunov exponent is positive is larger and larger. Considering molecular force, the clearance range with the positive maximum Lyapunov exponent is much larger than that of the case where molecular force is not considered. It shows that chaotic vibration is more likely to occur when the molecular force is considered.

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These results are because of the effects of the molecular force nonlinearity on system stiffness of the resonant sensor. The molecular force is also equivalent to a soft spring that causes decrease of the resonator stiffness. The molecular force nonlinearity causes changes of the resonator stiffness and more range of the initial clearance approaches chaotic vibration zone. So, the chaotic vibrations occur with considering molecular force, but do not occur without considering molecular force for some initial clearances. In a word, for small-clearance resonant sensor, the influence of molecular force on nonlinear vibration cannot be ignored.

15 * corresponding author: [email protected]

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0.1

with molecular force without molecular force =0

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w/m

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In order to verify the correctness of the theoretical analysis, the FEM dynamic displacement simulation of the micro-resonant sensor is carried out by using ANSYS analysis software. The finite element analysis model and grid partition of resonant sensor are given in figure 13a. The length of resonator is 2 mm, the initial tensile stress in x direction is 4.5╳106N/m2, the dynamic viscosity of gas is 1.86╳10-5N⋅S⋅m−2, and the excitation frequency is 31000 Hz. Changing the width of the resonator (from b=1.6mm to b=1.8mm), the dynamic displacements of the resonator are simulated. Figure 13b compares the maximum calculated displacements of the resonator with the FEM simulation results. Figs.13(c) and (d) show dynamic displacements in the resonator at b= 1.66mm and b=1.68mm, respectively. Results show: The FEM simulation results are close to the calculated results, with a maximum deviation of 9.77% and a minimum deviation of 4.23%. The simulated displacement is 1.109╳10-10m at b=1.66mm, and the simulated displacement is 0.878╳10-10m at b=1.68mm, here sudden change of the displacements of resonator occurs. For the width of b˂1.66mm, the displacements of resonator jumps greatly with changing the width b. For the width of b˃1.66mm, the displacements of resonator are stable. These simulation results are consistent with the bifurcation diagram corresponding to b=[1.6mm 1.8mm] in Fig.8 (a): There is a sudden change of the resonator displacements at b=1.66mm. When the width b increases from b=1.6mm to b=1.8mm, the vibrations of the resonator become from unstable chaotic vibration into stable periodic vibration.

1.1 1 0.9 0.8 1.6

(a) FEM model

1.7 b/mm

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(b) calculated and FEM results

16 * corresponding author: [email protected]

1.65

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(d) Displacements at b=1.68mm

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Fig. 13. FEM simulation of the dynamic displacements.

4. Conclusions

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A multi-field coupled dynamic model of a micro resonant pressure sensor was proposed in this paper. The effect of molecular force, electric field force, elastic force, and air damping force was considered, and the multi-field coupled dynamic equations of the resonator were provided. Using these equations, the nonlinear dynamical performance of the sensor was investigated. The results indicate the following: (1) For a given voltage, the initial clearance between the resonator and baseplate has an important effect on the vibration states of the micro resonant pressure sensor system as reducing the clearance could increase electrostatic nonlinearity. To avoid substandard dynamic performance of the resonant sensor, the initial clearance must be chosen properly. (2) For small clearance resonant sensors, when molecular force is considered, the amplitudes and range of the chaotic vibrations becomes large. This is due to the molecular force nonlinearity causing the resonator stiffness to change. (3) The operating pressure can influence vibration states of the sensor because the initial tension stress can alter the resonator stiffness. The resonator parameters should be adequately considered for a given pressure range in order to avoid these harmful vibration states. (4) The size of the resonator has an influence on both stiffness and mass of the resonator, which can be used to tune the vibration frequency of the sensor. (5) Small dynamic viscosity easily induces chaotic vibrations in the sensor. As the micro resonant pressure sensor operates at vacuum, the related parameters must be selected carefully to avoid harmful vibration states in the resonant sensor.

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References

[1] Zheng Y. Research progress and application prospect of smart sensor technology[J]. Science & Technology Review, 2016, 34(17):72-78. [2] Rafael N. G., Anna M. B., Alfons D. AC transfer function of electrostatic capacitive sensors based on 1-D equivalent model: application to silicon microphones. Journal of Microelectromechanical Systems, Vol. 12, Issue 6, 2003, p. 972-978. [3] Kumar D D, Kumar N, Kalaiselvam S, et al. Micro-tribo-mechanical properties of nanocrystalline TiN thin films for small scale device applications[J]. Tribology International, 2015, 88:25-30. [4] M.Gerhard, B.Sebastian, and P.Sumit, “Novel chemical sensor applications in commercial aircraft,” Procedia Engineering, Vol.25, 2011, pp.16-22. [5] Fu X, Xu L. Effects of Casimir force on multi-field coupled nonlinear vibration of orthotropic micro film[J]. Journal of Vibroengineering, 2017, 19(8). 17 * corresponding author: [email protected]

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[6] Lin W. Free vibration analysis of plane membranes by finite element method. Journal of Dynamics and Control, 2010. [7] Katsikadelis J T, Tsiatas G C. Nonlinear dynamic analysis of heterogeneous orthotropic membranes by the analog equation method[J]. Engineering Analysis with Boundary Elements, 2003, 27(2):115-124. [8] Fan S. Investigation on nonlinear vibration characteristics of resonant silicon microstructure pressure sensor[J]. Chinese Journal of Scientific Instrument, 2006, 27(12):1670-1673. [9] Reutskiy S Y. Vibration Analysis of Arbitrarily Shaped Membranes[J]. Computer Modeling in Engineering & Sciences, 2009, 51(2):115-142. [10] Li Q, Fan S, Tang Z, et al. Nonlinear vibration in resonant silicon bridge pressure sensor: Theory and experiment[C]// Solid-State Sensors, Actuators and Microsystems Conference. IEEE, 2011:1685-1688. [11] Tajaddodianfar F, Yazdi M R H, Pishkenari H N. Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method[J]. Microsystem Technologies, 2016:1-14. [12] Amorim T D, Dantas W G, Gusso A. Analysis of the chaotic regime of MEMS/NEMS fixed–fixed beam resonators using an improved 1DOF model[J]. Nonlinear Dynamics, 2014, 79(2):967-981. [13] Miandoab E M, Pishkenari H N, Yousefi-Koma A, et al. Chaos prediction in MEMS-NEMS resonators[J]. International Journal of Engineering Science, 2014, 82(3):74-83. [14] Shi H,Fan S,Xing W, et al. Study of Weak Vibrating Signal Detection Based on Chaotic Oscillator in Mems Resonant Beam Sensor[J]. Mechanical Systems and Signal Processing, 2015, 50–51: 535-547. [15] Alemansour H, Miandoab E M, Pishkenari H N. Effect of size on the chaotic behavior of nano resonators[J]. Communications in Nonlinear Science & Numerical Simulation, 2017, 44:495-505. [16] F, Pishkenari H N, Yazdi M R H. Prediction of chaos in electrostatically actuated arch micro-nano resonators: Analytical approach[J]. Communications in Nonlinear Science & Numerical Simulation, 2016, 30(1–3):182-195. [17] Bassinello D G, Tusset A M, Rocha R T, et al. Dynamical Analysis and Control of a Chaotic Microelectromechanical Resonator Model[J]. 2018, 2018(5). [18] Luo S, Li S, Tajaddodianfar F, et al. Observer-based adaptive stabilization of the fractional-order chaotic MEMS resonator[J]. Nonlinear Dynamics, 2018:1-11. [19]Tusset A M , Balthazar J M , Jeferson J D L , et al. On an Optimal Control Applied in Atomic Force Microscopy (AFM) Including Fractional-Order[J]. Proceedings of the ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference 2017:V004T09A003. Doi: https://doi.org/10.1115/DETC2017-67536 [20] Tusset A M , Balthazar J M , Bassinello D G , et al. Statements on chaos control designs, including a fractional order dynamical system, applied to a “MEMS” comb-drive actuator[J]. Nonlinear Dynamics, 2012, 69(4):1837-1857. Doi:https://doi.org/10.1007/s11071-012-0390-6 [21] Tusset A M, Bueno A M, Nascimento C B, et al. Nonlinear state estimation and control for chaos suppression in MEMS Resonator[J]. Shock & Vibration, 2014, 20(4):749-761. Doi: https://doi.org/10.3233/SAV-130782. [22] Pan J, Gu M. Geometric nonlinear effect to square tensioned membrane's free vibration[J]. Journal of Tongji University, 2007. [23] Zheng Z L, Liu C J, He X T, and Chen S L. Free Vibration Analysis of Rectangular Orthotropic Membranes in Large Deflection [J]. Mathematical Problems in Engineering, 2009: 634362. [24] Barauskas R, Kausinis S, Tilmans H A C. Investigation of Thermo-Elastic Damping of Vibrations of Rectangular and Ring-Shaped MEMS Resonators[J]. Journal of Vibroengineering, 2009, 11(1):177-187. [25] B. Geyer, G.L. Klimchitskaya, V.M. Mostepanenko,“Analytic approach to the thermal Casimir force between metaland dielectric,” Annals of Physics, Vol 323, pp.291-316, 2008. [26] Liu C J, Zheng Z L, He X T, et al. L-P Perturbation Solution of Nonlinear Free Vibration of 18 * corresponding author: [email protected]

ACCEPTED MANUSCRIPT Prestressed Orthotropic Membrane in Large Amplitude[J]. Mathematical Problems in Engineering, 2010, 2010(1024-123X):242-256. [27] Awrejcewicz J, Krys'Ko V A, Shitikova M V. Nonclassical Thermoelastic Problems in Nonlinear Dynamics of Shells: Applications of the Bubnov-Galerkin and Finite Difference Numerical Methods[J]. Applied Mechanics Reviews, 2012, 57(1):B6-B7. [28] Shin C, Kim W, Chung J. Free in-plane vibration of an axially moving membrane[J]. Journal of Sound & Vibration, 2004, 272(1):137-154.

Appendix A The stress in the resonator is proportional to the pressure on the film. It is illustrated as

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below:

The plane coordinate diagram of the pressure film is shown in figure A-1. If the thickness of the pressure film is d, the deflection of the pressure film is w, the boundary condition of the

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Fig.A-1 Plane coordinate diagram of the pressure film

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 w x  a  w y  b  0 f f  w w  0  x  y x  a y  b f f 

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Let the deflection w satisfying all boundary conditions be a trigonometric series     m x    n y   w   Amn 1  cos  1  cos     a    b    m 1 n 1 f f      

(A-2)

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In order to simplify the calculation, only take m  n  1 ，yields

   x     y  w  A11 1  cos  1  cos     a    b    f f      

The formula (B-3) is substituted into the Galerkin equation, yields









 



 





x y 1  cos   dxdy  0   P  D w 1  cos  a    b    4

Thus 19 * corresponding author: [email protected]

f

f



(A-3)

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4 Pa f 4  af 2 af 4   D  3  2 2  3 4  bf bf  

(A-4)

4

Here D

Ed 3 12 1   2 

(A-5)

x  

Ez   2 w 2w      1   2  x 2 y 2 

AN US

Ez   2 w 2w  y    2   1   2  y 2 x 

CR IP T

According to the relationship between deflection and stress, we can know

(A-6)

(A-7)

Substituting Eqs. (A-3) and (A-4) into (A-6) and (A-7), yields

 1 x  y  2 y  x   2 cos 1  cos   2 cos 1  cos  af  bf  bf bf  a f    a f

(A-8)

EzA11 2 y  1  2

 1 y  x  2 x  y   2 cos 1  cos   2 cos 1  cos  bf  af  af af  b f    b f

(A-9)

M

EzA11 2 1  2

x 

Letting y=0, we can obtain stress in the direction x as below

ED

EzA11 2  1  2

x

y 0

 2   x     2  2  cos  bf  a f b f 2   a f

(A-10)

PT

Letting x=0, we can obtain stress in the direction y as below

CE

y

x 0



EzA11 2 1  2

 2   y     2  2  cos  af  b f a f 2   b f

(A-11)

Because the clearance between the resonator and the pressure film is very small, the stress

AC

of the resonator can be approximated to that on the surface of the pressure film. Letting

z

d ，we obtain 2

x

y 0 z d

 2

24 Pa f 4  af 2  d  3  2 2 bf  2

2

 2   x    cos      2 b f 2  a f b f 2  a f 4   a f 3 4  b f 

20 * corresponding author: [email protected]

(A-12)

ACCEPTED MANUSCRIPT

x 0 z d

24 Pa f 4

 2



af 2



2

 2 d 2  3  2

bf

 2   y    cos      2 2 a f  b f a f 2  a f 4   b f 3 4  b f 

(A-13)

AC

CE

PT

ED

M

AN US

CR IP T

y

21 * corresponding author: [email protected]