Performance analysis of differential-frequency microgyroscopes made of nanocrystalline material

Accepted Manuscript

Performance Analysis of Differential-Frequency Microgyroscopes Made of Nanocrystalline Material M. Ghommem, A. Abdelkefi PII: DOI: Reference:

S0020-7403(16)30641-5 10.1016/j.ijmecsci.2017.09.008 MS 3919

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

31 October 2016 26 August 2017 8 September 2017

Please cite this article as: M. Ghommem, A. Abdelkefi, Performance Analysis of Differential-Frequency Microgyroscopes Made of Nanocrystalline Material, International Journal of Mechanical Sciences (2017), doi: 10.1016/j.ijmecsci.2017.09.008

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.

ACCEPTED MANUSCRIPT

Highlights • Modeling of vibrating beam microgyroscopes made of nanocrystalline material.

CR IP T

• Quasi-static and coupled dynamic analysis of rotating microbeams subject to electric actuation for gyroscopes operating in frequency mode. • Performance analysis of differential-frequency microgyroscopes.

AC

CE

PT

ED

M

AN US

• Design enhancement of microgyroscopes.

1

ACCEPTED MANUSCRIPT

Graphical Abstract

CR IP T

Increasing the grain size of the nanocrystalline material

Pull-in instability of the microgyroscope

Differential-frequency electrically–actuated microgyroscope made of nanocrystalline silicon

AN US

Increasing the grain size of the nanocrystalline material

AC

CE

PT

ED

M

Calibration curves of the microgyroscope

2

ACCEPTED MANUSCRIPT

Performance Analysis of Differential-Frequency Microgyroscopes Made of Nanocrystalline Material a

CR IP T

M. Ghommema and A. Abdelkefib

Department of Mechanical Engineering, American University of Sharjah, Sharjah 26666, UAE b

Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88003, USA

AN US

Abstract

AC

CE

PT

ED

M

The performance of a microgyroscope consisting of a rotating microbeam made of nanocrystalline silicon connected to a proof mass and subjected to electric actuation is investigated. The microgyroscope is assumed to operate in frequency-based mode; that is, the measurement of the rotation rate is extracted from the shift in the frequency response along the drive and sense directions. Closed-form expressions of the natural frequencies and mode shapes of the coupled dynamical system are derived. The onset of the base rotation is observed to split the common natural frequency of the two bending modes into a pair of closely-spaced natural frequencies. This frequency shift used as the output parameter detecting the rotation rate. A sensitivity analysis of this parameter to the rotation rate when varying the material properties of the microbeam and electric actuation is then performed. When applying the same DC voltage for the drive and sense modes, the differential frequency is found to vary linearly with the base rotation. Incorporating the fringing field in the electrostatic forcing and varying the grain size of the nanocrystalline silicon have insignificant impact on the calibration curve for this case. Breaking the symmetry of the microgyroscope in terms of the applied DC voltage along the drive and sense directions is observed to reduce significantly the sensitivity of the differential frequency to the base rotation rate. The larger the DC voltage bias is, the lower the sensor sensitivity is. Furthermore, under these operating conditions, the results show that the variations of the differential frequency with the rotation rate undergo nonlinear trends. Considering higher order modes is found to reduce the impact of the DC voltage bias on the calibration curve of the microgyroscope. Keywords: Microgyroscope, nanocrystalline silicon, differential frequency, electrostatic actuation, fringing field. Preprint submitted to International Journal of Mechanical Sciences

September 12, 2017

ACCEPTED MANUSCRIPT

1. Introduction

AC

CE

PT

ED

M

AN US

CR IP T

Inertial sensors detecting the position and orientation present crucial components of several navigation systems in air vehicles, automobiles, and satellites to track, and control their path. These include accelerometers used to measure displacements and gyroscopes deployed to determine rates of rotation about different axes. Recent advances in micromachining have enabled the fabrication of robust microgyroscopes at low cost with the capability to operate with high resolution and low power consumption [1, 2, 3]. Several studies have examined the performance of vibrating microgyroscopes with different designs including tunning forks, vibrating beams, and vibrating shells [3, 4, 5, 6, 7, 8, 9, 10]. The main purpose of these microsensors is to measure the angular velocity of a rotating frame through detecting the Coriolis force (inducing lateral motion) generated by the presence of combined rotational and transverse motions of a core element. The transverse motion (drive mode) can be achieved by the application of different actuation mechanisms, such as electrostatic, electromagnetic, and piezoelectric [11]. As for the lateral motion (sense mode), it is detected by deploying any of the transduction techniques (e.g., capacitive, piezoresistive, electromagnetic, optical...) in order to generate the output signal in response to a change in the applied rotation rates. The electrostatic approach presents the commonly-used technique for actuation and sensing of microsensors [11]. In this work, a microgyroscope consisting of a vibrating cantilever beam made of nanocrystalline silicon is considered. This microbeam is attached to a rotating base and connected to a rigid tip mass at the free end. The tip mass is coupled to two electrodes powered by a voltage source. One of these voltage sources is deployed to generate electrostatic force and excite the microbeam in the transverse direction and the other voltage source is utilized to detect the induced vibrations in the lateral direction, commonly converted to change in capacitance, and hence extract the base rotation rate. The operation of similar microsystems in amplitude-based mode has been previously analyzed in many studies [12, 13, 14, 15, 16, 17, 18, 19]. This approach relies on the magnitude of the induced vibrations to extract the rotation rate. In the present work, we follow a frequency-based approach [20] to measure the rate of rotation. We conduct static and dynamic analysis of the electrically-actuated microgyroscope under varying system configurations to provide guidance for enhancing the sensitivity of the sensing mechanism and achieving better performance. 4

ACCEPTED MANUSCRIPT

CE

PT

ED

M

AN US

CR IP T

The performance of microgyroscopes can be assessed at the earliest stages of design through the deployment of decent numerical models. Recent progress in computer hardware and software have enabled the numerical integration of the equations governing the performance of these systems [12, 13, 20, 21, 22, 23, 16, 17, 18, 19, 24, 14, 15]. These numerical models (i) enable the understanding of underlying physics and dynamics that govern these microsystems’ response, (ii) perform a rapid and reasonably accurate exploration of the available design space for performance optimization purposes, and (iii) test the potential feasibility of new sensing concepts and designs. In this work, we consider a mathematical model to analyze the performance of a microgyroscope whose main component is a vibrating microbeam made of nanocrystalline silicon under electric actuation and operating in frequency-based mode. We derive closed-form solutions of the quasi-static beam deflection when being subject to DC voltage and constant base rotation. The fringing field effect is considered and its impact on the pull-in instability is deeply studied. Operating away of this instability presents one of the design requirements for the sensor. The effects of the grain size of the nanocrystalline silicon on the microgyroscope static response, its natural frequencies, and its calibration curve are also examined. This effect is incorporated from a size-dependent micromechanical beam model developed by Shaat and Abdelkefi [25, 26] that provides an estimation of the effective material properties of nanocrystalline silicon with varying its grain size. The eigenvalue problem analysis shows that the base rotation couples the two global bending modes along the drive and sense directions and split their common natural frequency into a pair of closely-spaced natural frequencies. The difference between this pair of frequencies is considered as the output parameter detecting the rotation rate. The sensitivity of this parameter to the rotation rate when varying the grain size of the nanocrystalline silicon and electric actuation is carried out. The objective is to gain better understanding of microgyroscope made of nanocrystalline materials and come up with recommendations to optimize its design and performance.

AC

2. Modeling of a Microgyroscope Made of Nanocrystalline Material We consider a microgyroscope consisting of a microcantilever beam of length L, thickness h, width b, and uniform square cross section. The beam is attached to a rotating base, as shown in Figure 1. The microstructure is made of silicon as being the core material of electromechanical systems (MEMS) technology. A tip mass of finite size is connected to the free end of the beam and coupled to two electrodes for electric actuation. This actuation technique is a prevalent method 5

ACCEPTED MANUSCRIPT

Interface

𝑧

𝑦

Nano-grains

AN US

Rotating base

CR IP T

of driving MEMS devices because of its compatibility with microfabrication technology and capability to provide high power density [11]. We note that similar electrically-actuated microbeam systems have been proposed for microgyroscope applications based on the measurement of the beam vibrations along the sense direction via the capacitive transduction technique (amplitude-based mode) [12, 15, 14]. These previous works ignored the fringing field effect in the electrostatic forcing. The considered operating principle of the present microgyroscope is based on detecting the rotation rate from the shift in the natural frequency of the two vibrations modes (frequency-based mode).

𝑔𝑣

Ω

ℎ

𝑥

ℎ𝑀

𝑀

𝑏

𝑏𝑀

M

𝐿𝑀 𝑔𝑤

𝑉𝐷𝐶1

Sense electrode

𝑉𝐴𝐶

~

Drive electrode

PT

ED

𝑉𝐷𝐶2

CE

Figure 1: Schematic of the differential-frequency microgyroscope made of nanocrystalline silicon. It comprises a cantilever microbeam with a tip mass subject to electric actuation and under base rotational motion. The microgyroscope is assumed to operate in frequencybased mode; that is, the measurement of the rotation rate is extracted from the shift in the frequency response along the drive and sense directions.

AC

Microbeams are commonly made of nanostructured materials, such as nanocrystalline materials, nano-porous materials, and nanoparticle composites [25, 26]. Polycrystalline materials have two main phases, namely, grain and interface. In conventional coarse-grained polycrystalline materials, the amorphous interface phase is negligible. On the other hand, for nanocrystalline materials when grain sizes ranging between 1 nm and 100 nm, the amorphous interface phase can reach 40% [27]. Thus, unlike coarse-grained polycrystalline materials, nanocrystalline 6

ACCEPTED MANUSCRIPT

CR IP T

materials cannot be presented as homogeneous materials. In this work, the microstructure of the microgyroscope is made of nanocrystalline silicon and hence the effective properties of the structure are dependent on the size of the grain. To this end, following Shaat et al. [25, 28], an atomic lattice model and a two-phase micromechanical model, with the interface as the matrix material and the grains as nano-inclusions, are considered to estimate the effective material properties of beams made of nanocrystalline silicon for different grain size radii. In Table 1, the obtained effective Young’s modulus and density of the silicon for three cases of the grain average radius of 100 nm, 20 nm, and 0.5 nm are presented.

AN US

Table 1: The effective material properties of a nanocrystalline silicon with considering surface energy effects of the grain for different grain sizes [28].

Effective density (ρ) kg/m3 2291.1 2181.7 2152.2

M

Case Grain average radius Effective Young’s modulus (Rg ) nm (E) GPa 1 100 160.66 2 20 94.069 3 0.5 75.457

AC

CE

PT

ED

To determine the governing equations of motion of the microgyroscope shown in Figure 1, Euler-Bernoulli beam assumptions are considered. The microbeam undergoes two flexural displacements w(x, t) and v(x, t) along the transverse and lateral directions. This microbeam system is intended to sense the rotation rate. The operating principle of the sensor is based on the Coriolis effect. The microbeam is driven by an AC voltage in one direction (drive direction), which induces vibrations in an orthogonal direction due to the base rotation. The electrode placed along this direction (sense direction) is deployed to detect the induced motions and extract the underlying angular speed. Hereafter, ”0 ” denotes the first derivative with respect to x and ”. ” denotes the time derivative. After determining the expressions of the total kinetic and potential energies of the microstructure and applying the extended Hamilton’s principle, one obtains the following governing equations of motion [12, 13, 20]: ˙ − JΩ2 w00 − J Ωv ˙ 00 − J w¨ 00 = 0 (1) mw¨ + µw w˙ + EIwIV + 2mΩv˙ − mΩ2 w + mΩv

˙ − JΩ2 v 00 − J Ωw ˙ 00 − J v¨00 = 0 (2) m¨ v + µv v˙ + EIv IV − 2mΩw˙ − mΩ2 v − mΩw 7

ACCEPTED MANUSCRIPT

v = w = 0 and v 0 = w0 = 0 At x = L

AN US

v 00 = w00 = 0

CR IP T

where Ω denotes the base rotation rate, µ represents the damping coefficient, J is the mass moment of inertia, E is the effective Young’s modulus of the nanocrystalline silicon material, I is the beams cross sectional second moment of area, and m is the beam mass per unit length. The associated boundary conditions are given by: At x = 0 (3)

(4)

˙ − 2M Ωv˙ − M w¨ − J Ωv ˙ 0 − JΩ2 w0 − J w¨ 0 EIw000 + M Ω2 w − M Ωv   gw − w 1 0 Aw V 2 1 + βF (5) = − 2 (gw − w)2 w bM

M

˙ + 2M Ωw˙ − M v¨ − J Ωw ˙ 0 − JΩ2 v 0 − J v¨0 EIv 000 + M Ω2 v + M Ωw   gv − v 1 0 Av Vv2 1 + βF = − 2 2 (gv − v) hM

(6)

AC

CE

PT

ED

where gw and gv denote initial gap distances of the drive and sense capacitors, respectively. Aw = LM × bM and Av = LM × hM are the areas of the drive and sense capacitors, respectively. LM , bM , and hM denote the length, width, and thickness of the tip mass as shown in Figure 1. M represents the mass of the tip. The parameter 0 = 8.85 × 10−12 C2 N−1 m−2 is the permittivity of the dielectric vacuum between the tip mass and the electrode. Vw and Vv are the voltages applied along the drive and sense directions, respectively. Inspecting the boundary conditions, it is clear that the electrostatic force introduces nonlinearity. βF is a parameter introduced to study the impact of the fringing field on the operability of the microgyroscope. βF is set equal to zero when neglecting the effect of the fringing field and is set equal to 0.65 when considering this effect. Several studies of electrically-actuated micro-systems [29, 30, 31, 32, 33, 34, 35, 36, 37, 38] have demonstrated considerable effect of the fringing field and claimed the need to account for it in the modeling of electrostatically-actuated microbeams. Two models for incorporating the electric fringing field effects have been extensively used in the literature: the Palmer and the Mejis-Fokkema [33, 34, 35, 36, 37, 38]. In this work, we adopt 8

ACCEPTED MANUSCRIPT

xˆ =

CR IP T

the Palmer model to examine the impact of the fringing field effect, as being an inherent component of the electrostatic forcing, on the performance of the microgyroscope. The following parameters are used to nondimensionlize Equations (1)-(6) v w gv ˆ V ˆ Ω J x , vˆ = , wˆ = , δg = , t = τ t, Vˆ = , Ω = , Jˆ = , L gv gw gw Vref τ mL2 r 0 Av L3 2 M EI 0 Aw L3 2 ,τ = Vref , αv = Vref , ηm = , αw = 3 3 2EIgw 2EIgv mL mL4 µw µv bM ˆ hM µ ˆw = ,µ ˆv = , ˆbM = , hM = τ τ gw gv

(7)

AN US

and obtain the equations of motion in its nondimensional form as:

˙ − JΩ2 w00 − Jδg Ωv ˙ 00 − J w¨ 00 = 0 w¨ + µw w˙ + wIV + 2δg v˙ − Ω2 w + δg Ωv v¨ + µv v˙ + v IV − 2

(8)

1 ˙ 1 ˙ 00 1 Ωw˙ − Ω2 v − Ωw − JΩ2 v 00 − J Ωw − J v¨00 = 0 (9) δg δg δg

M

At x = 0, the dimensionless boundary conditions are given by:

At x = 1

ED

v = w = 0 and v 0 = w0 = 0

v 00 = w00 = 0

(10)

(11)

CE

PT

˙ − 2ηm δg Ωv˙ − ηm w¨ − Jδg Ωv ˙ 0 − JΩ2 w0 − J w¨ 0 w000 + ηm Ω2 w − ηm δg Ωv  αw 1 − w 2 = − V 1 + β (12) F (1 − w)2 w bM 1 1 ˙ 0 Ωw˙ − ηm v¨ − J Ωw − JΩ2 v 0 − J v¨0 δg δg  1 − v αv 2 = − V 1 + β (13) F v (1 − v)2 hM

AC

˙ + 2ηm v 000 + ηm Ω2 v + ηm Ωw

Note that the hats have been dropped for the sake of simpler notation. The above mathematical model is used to determine the influences of the grain size and fringing field on the performance of microgyroscopes made of nanocrystalline silicon. 9

ACCEPTED MANUSCRIPT

3. Performance Analysis for Design Enhancement of Microgyroscopes

AN US

CR IP T

3.1. Quasi-static analysis: effect of grain size, fringing field, base rotation on pull-in instability To examine the pull-in instability of the electrically-actuated beam system as being a critical issue for the operation and performance of the microgyroscope, the quasi-static problem is first solved. To this end, the time-dependent terms in the governing equations and boundary conditions are cancelled while the base rotation is kept constant. The resulting quasi-static equations for the drive and sense modes are uncoupled and hence they can be solved separately. For instance, the quasi-static problem of the rotating microbeam for the drive direction reduces to: wsIV − Ω2 ws − JΩ2 ws00 = 0

(14)

and subject to the following boundary conditions: ws (0) = ws0 (0) = 0 2

2

+ ηm Ω ws (1) − JΩ

ws00 (1) = 0

(15)

 2 1 − ws (1)  VDC1 1 + βF (16) = −αw (1 − ws (1))2 bw

M

ws000 (1)

and

ws0 (1)

ED

The general solution of equation (14) is given by [12]:

ws (x) = c1 ea1 x + c2 ea2 x + c3 ea3 x + c4 ea4 x

AC

CE

PT

where the coefficients ai ’s can be expressed as: p √ JΩ2 + Ω 4 + J 2 Ω2 √ a1,2 = ± 2 p √ 2 JΩ − Ω 4 + J 2 Ω2 √ a3,4 = ± 2

(17)

(18)

Using the given boundary conditions of the microgyroscope in Equation (15), the constants ci for i = {2, 3, 4} can be determined as function of the constant c1 and the general solution of Equation (14) can be expressed in terms of the coefficient c1 . Substituting this static solution into the nonlinear boundary condition given by Equation (16), a cubic equation in c1 can be obtained. Solving this equation leads to the evaluation of the static beam deflection under varying DC voltage and the 10

ACCEPTED MANUSCRIPT

CR IP T

identification of the pull-in instability onset which presents the upper limit of the electrical potential where the balance between the structural restoring force and electrostatic force is destroyed and the microbeam system collapses. The static analysis is crucial for the design of the microgyroscope as this sensor is expected to operate away from the pull-in instability. The geometrical and material properties of the present microgyroscope are summarized in Table 2. 1 0.9 0.8

AN US

0.7

𝛽𝐹 = 0 𝛽𝐹 = 0.65

0.5 0.4

Unstable

0.3

Stable

M

𝑤𝑠

0.6

0.2

1

2

3

Case 3 4

Case 2 5

Case 1 6

𝑉𝐷𝐶 (𝑉)

PT

0 0

ED

0.1

CE

Figure 2: Variations of the maximum static deflection with the applied DC voltage VDC for the drive direction when considering various cases of the grain size and with/without fringing field effect (Ω = 20 rad/s−1 ). The curves show stable (lower) and unstable (upper) branches. A decrease in the grain size of the nanocrystalline material is observed to reduce the pull-in voltage.

AC

We plot in Figure 2 the variations of the static deflection at the free end of the beam with the applied DC voltage VDC with and without considering the fringing field effects. Moreover, the shown results in Figure 2 are obtained for different material properties of the nanocrystalline silicon (cases 1, 2, and 3 shown in Table 1). Similar to the typical electrostatic actuators, the resulting curves, plotted in Figure 2, show stable (lower) and unstable (upper) branches. It follows from the plotted curves in Figure 2 that a decrease in the grain size of the nanocrystalline material reduces the pull-in voltage. This is expected due to the softening effect 11

ACCEPTED MANUSCRIPT

Table 2: Microgyroscope parameters [12].

CR IP T

Numerical values 400 2.8 2.8 see Table 1 see Table 1 ρ×b×h 0.5 m × L (192,192) (2,2) (4 gw , 4 gv )

AN US

Parameter Beam length (L) µm Beam thickness (h) µm Beam width (b) µm Density (ρ) kg/m3 Young’s modulus (E) N/m2 Mass per unit length (m) kg/m Tip mass (M ) kg Capacitor areas (Aw , Av ) µm2 Intial gap distance (gw , gv ) µm Tip mass dimensions (bM , hM ) µm

AC

CE

PT

ED

M

associated with the decrease in the grain size. Indeed, the effective Young’s modulus of the nanocrystalline silicon decreases when the grain size is decreased, as presented in Table 1. To this end, a lower DC voltage is needed to reach pull-in. As for the effects of the fringing field on the pull-in instability of the driving mode, it is clear that the fringing field affects the static pull-in of the microgyroscope. Indeed, the fringing field increases the contribution of the electrostatic force and hence lower values in the pull-in voltage are obtained, as presented in Figure 2. Incorporating the fringing field impact in the electric actuation yields further reduction of the pull-in voltage VP I . A 5-6% decrease in VP I was observed for the three cases under study. In Figure 3, the effect of the base rotation rate Ω on the pull-in voltage VP I is showed when considering various grain sizes of the nanocrystalline silicon and with and without considering the impact of the fringing field. Inspecting the plotted curves in Figure 3, it is noted that an increase in the base rotation rate Ω results in a slight decrease in the pull-in voltage. This is right when the base rotation rate is less than 104 rad/s. However, when Ω reaches very high values (in the order of 104 rad/s), its impact on the pull-in voltage becomes significant and its increase yields a sharp drop in the pull-in voltage. A very high angular rotation is not commonly encountered, however, the results show that if such situation occurs, one needs to account for its effect on the stability of the system in the presence of the electrostatic actuation. All simulated cases exhibit similar trends in the variation of VP I with Ω. Furthermore, it is observed that the effect of the fringing field on 12

ACCEPTED MANUSCRIPT

the pull-in voltage reduces at high rotation rates. 6

𝛽𝐹 = 0

CR IP T

Case 1

5.5

𝛽𝐹 = 0.65

5

Case 2 Case 3

4 3.5 3 2.5

10

1

10

2

10

Ω (rad/s)

3

10

4

10

5

M

2 0 10

AN US

𝑉𝑃𝐼 (𝑉)

4.5

ED

Figure 3: Variations of the pull-in voltage VP I as a function of the base rotation rate Ω when considering various cases of the grain size and with/without fringing field effect. An increase in the base rotation rate Ω results in a slight decrease in the pull-in voltage.

AC

CE

PT

It can be concluded from this quasi-static analysis that the fringing field and the decrease in grain size strongly affect the static position and pull-in instability of the sense and drive modes. On the other hand, for realistic values of the base rotation rate, it is noted that it has a negligible effect on the pull-in instability of the microgyroscope. Therefore, this rotation rate cannot be detected from a quasistatic or static analyses as performed in this section. Next, the coupled frequencies of the sense and drive modes are determined with including the effects of the base rotation rate, fringing field, and grain size of the nanocrystalline silicon. 3.2. Coupled dynamic analysis As mentioned above, the main purpose of gyroscopes is to detect the base rotation rate as needed to track the orientation of a moving dynamical system. In an amplitude modulation electrically-actuated gyroscope, the magnitude of the vibrations in the sense direction, inducing a capacitive change between the sense 13

ACCEPTED MANUSCRIPT

AN US

CR IP T

electrode and the tip mass, is employed to measure the angular rotation speed. Nayfeh et al. [20] proposed to extract the base rotation rate from the differential frequency between the drive and sense modes. We assume that the operation of the present microgyroscope follows this frequency-based approach where the performance of the sensing system is associated with the sensitivity of the differential frequency to the change in the angular speed of the rotating frame. Note that this frequency-based method for sensing the rotation rate can be employed for both free and forced vibrations of the microbeam. In this section, the impacts of the effective material properties and fringing field on the coupled frequencies of the microgyroscope are investigated in order to come up with recommendations to improve the operability of microgyroscopes based on vibrating beams. To this end, following Ghommem et al. [12], the coupled natural frequencies and mode shapes of the dynamical system are evaluated. The microbeam response is decomposed into static and dynamic components as: w(x, t) = ws (x) + (Φw (x)eiωt + cc) v(x, t) = vs (x) + (Φv (x)eiωt + cc)

(19)

PT

ED

M

where (Φw , Φv ) represent, respectively, the mode shapes of the microbeam in the drive and sense directions, ω is the coupled natural frequency, and cc stands for the complex conjugate of the preceding term. Substituting Equations (19) into Equations (8)-(9), dropping the damping terms, considering a constant base rotation rate Ω, and expanding the nonlinear electrostatic forcing term in Taylor series in equations (12)-(13) while keeping only the static and linear term of the expansion, the following system of equations is obtained:

  1 0 A1 = , 0 1

CE

where   Φw Φ= , Φv

A1 Φiv + A2 Φ00 − A3 Φ = 0

2

(20)

2

A2 = −(Jω + 2JΩ )A1 ,

  2 Ω + ω 2 2iδg Ωω (21) A3 = Ω2 + ω 2 − 2iΩω δg

AC

The associated boundary conditions are given by: At x = 0 Φv = Φw = 0 and Φ0v = Φ0w = 0

(22)

Φ00v = Φ00w = 0

(23)

At x = 1

14

ACCEPTED MANUSCRIPT

(24)

CR IP T

2 2 2 0 2 0 Φ000 w + ηm Ω Φw − 2iηm δg ΩωΦv + ηm ω Φw − JΩ Φw − Jω Φw   2 1 1 2 = −αw VDC1 Φw + β F (1 − ws )3 bw (1 − ws )2

1 ΩωΦw + ηm ω 2 Φv − JΩ2 Φ0v − Jω 2 Φ0v δg   2 1 1 2 = −αv VDC1 Φv + β F (1 − ws )3 bv (1 − ws )2

+ ηm Ω2 Φv + 2iηm Φ000 v

AN US

The spatial functions Φw and Φv are expressed in the following form:   Φw,i si x Φ= e Φv,i

(25)

(26)

M

Substituting Equation (26) into Equation (20), we obtain the following system of equations h i Φ  0 w,i 4 2 si A1 + si A2 − A3 = (27) Φv,i 0

where

PT

ED

solution of Equation (27)), the determinant of the matrix h To obtain a nontrivial i 4 2 si A1 + si A2 − A3 is set equal to zero. This yields the general solution which can be expressed as: Φ(x) =

 8  X Φw,i i=1

Φv,i

esi x

(28)

AC

CE

p √ Jω 2 + JΩ2 − 4ω 2 + J 2 ω 4 − 8ωΩ + 4ω 2 + 2J 2 ω 2 Ω2 + J 2 Ω2 √ s1,2 = ± 2 p √ Jω 2 + JΩ2 − 4ω 2 + J 2 ω 4 + 8ωΩ + 4ω 2 + 2J 2 ω 2 Ω2 + J 2 Ω2 √ s3,4 = ± 2 r 2 2 Jω JΩ 1p s5,6 = ± + + −4(−ω 2 − 2ωΩ − Ω2 ) + (Jω 2 + JΩ2 )2 2 2 2 r Jω 2 JΩ2 1 p s7,8 = ± + + −4(−ω 2 + 2ωΩ − Ω2 ) + (Jω 2 + JΩ2 )2 (29) 2 2 2 15

ACCEPTED MANUSCRIPT

and Φw,i =



−Φv,i s4i

−

s2i (Jω 2

  2iωΩ  + JΩ ) − (ω + Ω ) / − δg 2

2

2

(30)

M (ω) Φv,1 Φv,2 ... Φv,8 .


T

CR IP T

Next, we substitute the spatial solution given by Equation (28) into the boundary conditions (22)-(25) to obtain a linear homogeneous algebraic system which can be written in the following form: =0

(31)

ED

M

AN US

M (ω) is 8 × 8 matrix where the coefficients are function of the natural frequency ω. Equation (31) has a nontrivial solution if and only if the determinant of the matrix M (ω) vanishes. Solving the nonlinear equation obtained by imposing this condition, the coupled natural frequencies of the dynamical system can be obtained. In Table 3, the first three dimensional natural frequencies of the microgyroscope for both sense and drive modes when VDC = 0 and Ω = 0 are presented when considering different grain size of the nanocrystalline silicon. Clearly, the smaller the grain size of the nanocrystalline silicon used for the microbeam is, the lower the natural frequency is. This is expected due to the fact that a decrease in the grain size is accompanied with a decrease in the effective Young’s modulus of the structure and hence its natural frequency.

Case 1 2 3

ω1 (kHz) 84.961 65.121 58.169

ω2 (kHz) 712.179 545.875 487.596

AC

CE

PT

Table 3: The first three dimensional natural frequencies of the microbeam for different effective material properties.

ω3 (kHz) 2178.530 1669.820 1491.540

The onset of the base rotation splits the common natural frequency of the drive and sense modes into two-closely spaced frequencies: ωd and ωs . We plot in Figure 4 the variations of the first coupled natural frequency (dimensionless) of the drive bending mode with the applied DC voltage for different grain size of the nanocrystalline silicon and with/without considering the effect of the fringing field. As the DC voltage is increased, the coupled natural frequency of the drive 16

ACCEPTED MANUSCRIPT

CR IP T

mode decreases and undergoes a sharp drop in the vicinity of the pull-in instability. At the onset of the pull-in, the coupled natural frequency reaches zero as the tip mass collapses on the electrode. All cases show similar trend in the variations of the coupled natural frequency with the DC voltage. As for the impacts of the fringing field on the coupled natural frequency of the drive mode, it is noted that a slight influence is obtained due to the increase in the electrostatic effect and hence a decrease in the needed DC voltage. When VDC approaches the pull-in voltage, the effect of the fringing field on the coupled natural frequency of the microgyroscope becomes more significant.

AN US

2.5

𝛽𝐹 = 0

2

𝛽𝐹 = 0.65

M

𝜔1𝑑

1.5

0.5

1

2

Case 2

Case 3

Case 1 3

4

5

6

𝑉𝐷𝐶 (𝑉)

CE

PT

0 0

ED

1

AC

Figure 4: Variations of the first coupled natural frequency ω1d for the drive direction when considering various cases of the grain size and with/without fringing field effect (Ω = 20 rad/s). The natural frequency decreases when the DC voltage is increased and undergoes a sharp drop in the vicinity of the pull-in instability.

Next, we define the dimensional differential frequency as follows: h i ∆˜ ω = |ωd − ωs | − |ωd − ωs |Ω=0 × τ

(32)

where ωd and ωs are the natural frequencies associated with the drive and sense modes. Note that we have ωd = ωs when the same DC voltage is applied along 17

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

the drive and sense directions and the base rotation rate Ω is set equal to zero. The term |ωd − ωs |Ω=0 is the difference in the natural frequencies between the drive and sense modes associated with the bias DC voltages (VDC1 6= VDC2 ) and the parameter τ is defined in Equation (7). To operate in the frequency-based mode, the differential frequency ∆˜ ω presents the basis to estimate the base rotation and is taken as the output of the sensor. The sensitivity of ∆˜ ω to the base rotation rate is examined when considering the effects of the grain size of the nanocrystalline silicon and fringing field. In Figure 5, the calibration curves of the microgyroscope for different electrostatic actuations, with and without including the fringing field is presented when considering a grain size for the nanocrystalline silicon equals to 100 nm. This curve shows the variations of the differential frequency ∆˜ ω with the base rotation rate Ω. The results are obtained for a full span of 200 rad/s. When applying the same DC voltage along the drive and sense directions (i.e., VDC1 = VDC2 ), the differential frequency ∆˜ ω varies linearly with the base rotation, as demonstrated by Nayfeh et al. [20] when using perturbation techniques. The sensitivity of the microgyroscope is found equal to 2 Hz/(rad/s). Of interest, the fringing field has no impact on the calibration curve in this case. This indicates that capturing the shift in the coupled natural frequencies from the free vibrations of the microbeam can produce a practical measure of the angular speed. Breaking the symmetry of the electrically-actuated gyroscope in terms of the applied DC voltage along the drive and sense directions (i.e., VDC1 6= VDC2 ) is observed to reduce significantly the sensitivity of the differential frequency to the base rotation rate. The larger the voltage bias (|VDC2 −VDC1 |) is, the lower the sensor sensitivity is. For instance, the sensitivity reduces to less than 0.25 Hz/(rad/s) for |VDC2 −VDC1 | = 1.5 V. Under these operating conditions, the calibration curves show nonlinear trends. Furthermore, the Fringing field effect reduces further the sensitivity of the microgyroscope. The results demonstrate the design requirement to consider applying the same electrostatic actuation for both drive and sense directions to achieve better performance of the microgyroscope. Next, the impact of the effective material properties of the nanocrystalline silicon used for the microbeam on the overall performance of the microgyroscope is investigated. In Figures 6(a) and 6(b), we show the calibration curves obtained for varying material properties (cases 1, 2, and 3 shown in Table 1) and DC voltage bias levels, with and without considering the fringing field effect. The results show insignificant impact of the grain’s size of the nanocrystalline material on the differential frequency ∆˜ ω as long as the same DC voltage is applied in the drive and sense directions. The coupled dimensional first natural frequencies of the 18

ACCEPTED MANUSCRIPT

400 𝛽𝐹 = 0.65

350

Δ𝜔 (Hz)

250

CR IP T

𝛽𝐹 = 0

300

𝛽𝐹 = 0.65

200 𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0

𝛽𝐹 = 0

150 𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0.5 V

AN US

100

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 1.5 V

50 0 0

50

100

150

200

Ω (rad/s)

PT

ED

M

Figure 5: Calibration curve: variations of the frequency shift ∆˜ ω as a function of the base angular velocity Ω in dimensional form with/without considering the fringing field when the grain size for the nanocrystalline silicon is equal to 100 nm. The sensitivity of the microgyroscope is found equal to 2 Hz/(rad/s). The fringing field has no impact on the calibration curve in this case. Breaking the symmetry of the electrically-actuated gyroscope in terms of the applied DC voltage degrades the performance of the microgyroscope. The larger the voltage bias (|VDC2 − VDC1 |) is, the lower the sensor sensitivity is.

AC

CE

drive and sense modes and the associated frequency shift when considering three cases of the grain size of the nanocrystalline silicon while applying the same DC voltage along the drive and sense directions are shown in Tables 4 and 5. A negligible impact in the frequency shift is observed when accounting for the fringing field. Introducing a DC voltage bias leads to a noticeable effect on the calibration curves in terms of linearity and slope. Moreover, considering nanocrystalline silicon with smaller grain size (resulting in smaller Young’s modulus) reduces the sensitivity of the differential frequency to the base rotation rate and then affects the operability of the microgyroscope, as shown in Figures 6(a) and 6(b). For instance, when operating at a DC voltage bias of |VDC2 − VDC1 | = 1.5 V, the differential frequency ∆˜ ω is insensitive to any rotation rate smaller than 50 rad/s 19

ACCEPTED MANUSCRIPT

Table 4: Dimensional natural frequencies of the microbeam for different effective material properties for VDC1 = VDC2 = 1V (without fringing field effect).

Case 2

Case 3

CR IP T

Case 1

AN US

Ω = 0 Ω = 100 Ω = 0 Ω = 100 Ω = 0 Ω = 100 (rad/s) (rad/s) (rad/s) ω1d (Hz) 84,483 84,583 64,489 64,589 57,454 57,554 ω1s (Hz) 84,483 84,383 64,489 64,389 57,454 57,354 ∆˜ ω (Hz) 0 200 0 200 0 200

Table 5: Dimensional natural frequencies of the microbeam for different effective material properties for VDC1 = VDC2 = 1V (with fringing field effect).

Case 1

Case 2

Case 3

ED

M

Ω = 0 Ω = 100 Ω = 0 Ω = 100 Ω = 0 Ω = 100 (rad/s) (rad/s) (rad/s) ω1d (Hz) 84,443 84,543 64,434 64,534 57,392 57,492 ω1s (Hz) 84,443 84,343 64,434 64,334 57,392 57,292 ∆˜ ω (Hz) 0 200 0 200 0 200

AC

CE

PT

and then this affects the performance of the microgyroscope. In Tables 6, 7, 8, and 9, we present the coupled dimensional first natural frequencies of the drive and sense modes when considering asymmetry in the applied DC voltages while varying the grain sizes of the nanocrystalline silicon with/without including the fringing field. The values of the natural frequencies are computed for Ω = 0 rad/s and 100 rad/s. Clearly, a microgyroscope (operating in the frequency mode) made of smaller grain sizes has less sensitivity to changes in the base angular velocity. We examine the performance of the frequency-based microgyroscope operating at higher order modes. We plot in Figure 7 the variations of the differential frequency with the base rotation rate when operating around the second natural frequency. The results are obtained for varying nanocrystalline material properties. Applying the same DC voltage yields the same sensitivity of ∆˜ ω to the rotation rate as that obtained when operating around the first natural frequency.

20

ACCEPTED MANUSCRIPT

400

400

350 300

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0

Δ𝜔 (Hz)

Δ𝜔 (Hz)

300 250 200 150

250

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0

200 150

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0.5 V

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0.5 V

100

100 𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 1.5 V

50

50

100

Ω (rad/s)

150

(a) Without Fringing field

200

0 0

50

100

Ω (rad/s)

M

CE

PT

ED

However, the DC voltage bias is observed to have less impact on the calibration curves in terms of linearity and slope in comparison with the operation of the microgyroscope at the first mode. Furthermore, the fringing field shows insignificant effect on the calibration curves for all simulated cases. These results suggest the need to operate the microgyroscope at higher order modes in the presence of DC voltage bias. 3.3. Design specifications Based on the performed coupled dynamical analysis, the following recommendations are proposed in order to enhance the overall performance of microgyroscopes made of nanocrystalline silicon.

AC

150

(b) With Fringing field

Figure 6: Variations of the frequency shift ∆˜ ω as a function of the base angular velocity Ω in dimensional form for various grain sizes of the naocrystalline silicon and two distinct asymmetric DC voltages: (a) without and (b) with considering the impact of the fringing field. Insignificant impact of the grain’s size of the nanocrystalline material is observed on the differential frequency ∆˜ ω . Introducing a DC voltage bias leads to a noticeable effect on the calibration curves of the microgyroscope in terms of linearity and slope.

• Design microgyroscopes made of nanocrystalline silicon with large grain size to shift the pull-in voltage to higher values and then enlarge the operation range of the electrically-actuated microgyroscope.

21

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 1.5 V

50

AN US

0 0

Case 1 Case 2 Case 3

CR IP T

350

Case 1 Case 2 Case 3

200

ACCEPTED MANUSCRIPT

Table 6: Dimensional natural frequencies of the microbeam for different effective material properties for VDC2 − VDC1 = 0.5V (without fringing field effect).

Case 2

Case 3

CR IP T

Case 1

AN US

Ω = 0 Ω = 100 Ω = 0 Ω = 100 Ω = 0 Ω = 100 (rad/s) (rad/s) (rad/s) ω1d (Hz) 84,842 84,868 64,964 64,984 57,991 58,010 ω1s (Hz) 84,489 84,463 64,503 64,482 57,474 57,455 ∆˜ ω (Hz) 0 52 0 41.6 0 38

Table 7: Dimensional natural frequencies of the microbeam for different effective material properties for VDC2 − VDC1 = 0.5V (with fringing field effect).

Case 1

Case 2

Case 3

ED

M

Ω = 0 Ω = 100 Ω = 0 Ω = 100 Ω = 0 Ω = 100 (rad/s) (rad/s) (rad/s) ω1d (Hz) 84,832 84,857 64,950.6 64,969.9 57,975.9 57,993.2 ω1s (Hz) 84,450 84,426 64,451.8 64,432.5 57,416.6 57,399.3 ∆˜ ω (Hz) 0 49 0 38.6 0 34.6

CE

PT

• Operate the microgyroscope in the frequency mode which produces easy detection of the rotation rate (linear behavior). The output parameter is insensitive of the fringing field effect and effective material properties when the same DC voltage is applied for both drive and sense modes. • Avoid applying DC voltage bias that affects the linearity of the calibration curve and reduces the sensitivity of the microgyroscope.

AC

• Consider operating the microgyroscope at higher order modes in the presence of DC voltage bias. Less impact of the DC voltage bias on the sensitivity of ∆˜ ω to the variations in the base rotation rate is obtained when considering higher order modes.

We note that the shift in the natural frequency between the two vibration modes of the present microgyroscope resulting from low base rotation 22

ACCEPTED MANUSCRIPT

Table 8: Dimensional natural frequencies of the microbeam for different effective material properties for VDC2 − VDC1 = 1.5V (without fringing field effect).

Case 2

Case 3

CR IP T

Case 1

AN US

Ω = 0 Ω = 100 Ω = 0 Ω = 100 Ω = 0 Ω = 100 (rad/s) (rad/s) (rad/s) ω1d (Hz) 84,835.5 84,841.1 64,948 64,952.4 57,967.5 57,971.3 ω1s (Hz) 83,057.8 83,052.1 62,610 62,605.8 55,337.4 55,333.6 ∆˜ ω (Hz) 0 11.3 0 8.6 0 7.6

Table 9: Dimensional natural frequencies of the microbeam for different effective material properties for VDC2 − VDC1 = 1.5V (with fringing field effect).

Case 1

Case 3

M

Ω = 0 Ω = 100 Ω = 0 Ω = 100 Ω = 0 Ω = 100 (rad/s) (rad/s) (rad/s) 84,823.9 84,829.1 64,930.5 64,934.4 57,945.6 57,949.1 82,900 82,894 62,398.8 62,394.9 55,097.1 55,093.6 0 10.2 0 7.8 0 7.0

ED

ω1d (Hz) ω1s (Hz) ∆˜ ω (Hz)

Case 2

PT

rates is small. The highest sensitivity achieved in this numerical analysis is 2 Hz/rad/s. As such, the microgyroscopes relying on the frequency-based approach are more suitable for detecting high angular speeds.

CE

4. Conclusions

AC

The performance of an electrically-actuated microgyroscope consisting of a micro cantilever beam made of nanocrystalline silicon attached to a rotating base and connected to a proof mass was investigated. A quasi-static analysis was first carried out to determine the pull-in instability of the microbeam when subjected to various rotation rates while considering different grain sizes of the nanocrystalline silicon and including the fringing field effect. Closed-form expressions of the coupled natural frequencies and mode shapes of the coupled system were derived through an eigenvalue problem analysis. The onset of the base rotation was 23

ACCEPTED MANUSCRIPT

CR IP T

observed to shift the coupled natural frequencies of the two bending modes (drive and sense modes). The shift in the coupled frequencies was used as the detector of the base rotation. Calibration curves of the sensor were obtained for varying electric actuation and grain size of the nanocrystalline silicon. Introducing DC voltage bias was observed to result in a noticeable deterioration of the overall performance of the microgyroscope made of nanocrystalline silicon. However, less impact of DC voltage bias on the calibration curves was obtained when operating the microgyroscope around the second natural frequency. References

AN US

References

[1] N. Yazdi, F. Ayazi, K. Najafi, Micromachined inertial sensors, IEEE 86 (1996) 1640–1659. [2] C. B. Williams, C. Shearwood, P. H. Mellor, A. D. Mattingley, M. R. Gibbs, R. B. Yates, Initial fabrication of a micro-induction gyroscope, Microelectronic Engineering 30 (1996) 531–534.

ED

M

[3] S. Sassen, R. Voss, J. Schalk, E. Stenzel, T. Gleissner, R. Gruenberger, F. Neubauer, W. Ficker, W. Kupke, K. Bauer, M. Rose, Tuning fork silicon angular rate sensor with enhanced performance for automative applications, Sensors and Actuators 83 (2000) 80–86.

PT

[4] M. Kurosawa, Y. Fukuda, M. Tkasaki, T. Huguch, A surface-acoustic-wave gyro sensor, Sensors and Actuators 66 (1998) 33–39.

CE

[5] K. Maenaka, T. Fujita, Y. Konishi, M. Maeda, Analysis of a highly sensitive silicon gyroscope with cantilever beam as vibrating mass, Sensors and Actuators 54 (1996) 568–573.

AC

[6] S. Mohite, N. Patil, R. Pratap, Design, modeling and simulation of vibratory micromachined gyroscopes, Journal of Physics 34 (2006) 757–763. [7] C. Acar, A. M. Shkel, Structural design and experimental characterization of torsional micromachined gyroscopes with non-resonant drive mode, Journal of micromechanics and microengineering 14 (2004) 15–25. [8] E. Alper, T. Akin, A symetric surface micromachined gyroscope with decoupled oscillation modes, Sensors and Actuators 97-98 (2004) 347–358. 24

ACCEPTED MANUSCRIPT

[9] Y. S. Hong, J. H. Lee, S. H. Kim, A lateraly driven symmetric micro resonator for gyroscopic applications, Journal of Micromachining and Microengineering 10 (2000) 452–458.

CR IP T

[10] C. Jeong, A study on resonant frequency and q factor tunings for mems vibratory gyroscopes, Journal of micromechanics and microengineering 14 (2004) 1530–1536.

[11] M. I. Younis, MEMS linear and nonlinear statics and dynamics, Springer, 2011.

AN US

[12] M. Ghommem, A. Nayfeh, S. Choura, F. Najar, E. Abdel-Rahman, Modeling and performance study of a beam microgyroscope, Journal of Sound and Vibration 329 (2010) 4970–4979. [13] M. Ghommem, A. Nayfeh, S. Choura, Model reduction and analysis of a vibrating beam microgyroscope, Journal of Vibration and Control 19 (2013) 1240–1249.

M

[14] M. Esmaeili, N. Jalili, M. Durali, Dynamic modeling and performance evaluation of a vibrating beam microgyroscope under general support motion, Journal of Sound and Vibration 301 (2006) 146–164.

ED

[15] V. Bhadbhade, N. Jalili, S. N. Mahmoodi, A novel piezoelectrically actuated flexural/torsional vibrating beam gyroscope, Journal of Sound and Vibration 311 (2008) 1305–1324.

CE

PT

[16] M. Mojahedi, M. T. Ahmadian, , K. Firoozbakhsh, Oscillatory behavior of an electrostatically actuated microcantilever gyroscope, International Journal of Structural Stability and Dynamics 13 (2013) 1350030(24 pages).

AC

[17] M. Mojahedi, M. T. Ahmadian, , K. Firoozbakhsh, The oscillatory behavior, static and dynamic analyses of a micro/nano gyroscope considering geometric nonlinearities and intermolecular forces, Acta Mechanica Sinica 29 (2013) 851–863. [18] M. Mojahedi, M. T. Ahmadian, , K. Firoozbakhsh, The influence of the intermolecular surface forces on the static deflection and pull-in instability of the micro/nano cantilever gyroscopes, Composites: Part B 56 (2014) 336–343.

25

ACCEPTED MANUSCRIPT

[19] M. Rasekh, S. Khadem, Design and performance analysis of a nanogyroscope based on electrostatic actuation and capacitive sensing, Journal of Sound and Vibration 332 (2013) 6155–6168.

CR IP T

[20] A. Nayfeh, E. Abdel-Rahman, M. Ghommem, A novel differential frequency micro-gyroscope, Journal of Vibration and Control 21 (2015) 872–882. [21] M. H. Ghayesha, H. Farokhib, G. Alici, Size-dependent performance of microgyroscopes, International Journal of Engineering Science 100 (2016) 99– 111.

AN US

[22] S. A. M. Lajimi, G. R. Heppler, E. M. Abdel-Rahman, Primary resonanceofabeam-rigidbodymicrogyroscope, International Journal of NonLinear Mechanics 77 (2015) 364–375. [23] S. A. M. Lajimi, G. R. Heppler, E. M. Abdel-Rahman, A mechanicalthermal noise analysis of a nonlinear microgyroscope, Mechanical Systems and Signal Processing 83 (2017) 163–175.

M

[24] Y. Li, S. Fan, Z. Guo, J. Li, L. Cao, Frequency measurement study of resonant vibratory gyroscopes, Journal of Sound and Vibration 331 (2012) 44174424.

PT

ED

[25] M. Shaat, A. Abdelkefi, Modeling the material structure and couple stress effects of nanocrystalline silicon beams for pull-in and bio-mass sensing applications, International Journal of Mechanical Sciences 101-102 (2015) 280–291.

CE

[26] M. Shaat, A. Abdelkefi, Pull-in instability of multi-phase nanocrystalline silicon beams under distributed electrostatic force, International Journal of Engineering Science 90 (2015) 58–75.

AC

[27] S.-W. Y. G-F. Wang, X-Q Feng, C.-W. Nan, Interface effects on effective elastic moduli of nanocrystalline materials, Materials Science and Engineering A 363 (2003) 1–8. [28] M. Shaat, M. A. Khorshidi, A. Abdelkefi, M. Shariati, Modeling and vibration characteristics of cracked nano-beams made of nanocrystalline materials, International Journal of Mechanical Sciences 115-116 (2016) 574–585.

26

ACCEPTED MANUSCRIPT

CR IP T

[29] A. Farrokhabadia, A. Mohebshahedinb, R. Rachc, J.-S. Duand, An improved model for the cantilever nems actuator including the surface energy, fringing field and casimir effects, Physica E: Low-dimensional Systems and Nanostructures 75 (2016) 202–209. [30] Y. Sun, Y. Yu, B. Wu, B. Liu, Closed form solutions for nonlinear static response of curled cantilever micro-/nanobeams including both the fringing field and van der waals force effect, Microsystem Technologies (2016) 1–12.

AN US

[31] K. Guha, M. Kumar, S. Agarwal, S. Baishya, A modified capacitance model of rf mems shunt switch incorporating fringing field effects of perforated beam, Solid-State Electronics 114 (2016) 35–42. [32] L. X. Zhang, Y.-P. Zhao, Electromechanical model of rf mems switches, Microsystem Technologies 9 (2003) 420–426. [33] R. C. Batra, M. Porfiri, D. Spinello, Capacitance estimate for electrostatically actuated narrow microbeams, Micro & Nano Letters 1 (2006) 7173.

M

[34] R. C. Batra, M. Porfiri, D. Spinello, Electromechanical model of electrically actuated narrow microbeams, Journal of Microelectromechanical Systems 15 (2006) 11751189.

ED

[35] R. C. Batra, M. Porfiri, D. Spinello, Vibrations of narrow microbeams predeformed by an electric field, Journal of Sound and Vibration 309 (2008) 600–612.

CE

PT

[36] K. Das, R. C. Batra, Pull-in and snap-through instabilities in transient deformations of microelectromechanical systems, Journal of Micromechanics and Microengineering 19 (2009) 035008035027.

AC

[37] P.-P. Chao, C. W. Chiu, T.-H. Liu, Dc dynamic pull-in predictions for a generalized clampedclamped micro-beam based on a continuous model and bifurcation analysis, Journal of Micromechanics and Microengineering 18 (2008) 115008. [38] H. M. Ouakad, Electrostatic fringing-fields effects on the structural behavior of mems shallow arches, Microsystem Technologies (2016) 1–9.

27

ACCEPTED MANUSCRIPT

400

400

210

210

205

200

200

350

195 190

160

180

300

175 170

100

105

110

250

115

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 1.5 V

150

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0.5 V

100

200

80

90

100

110

120

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0

150

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 1.5 V

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0.5 V

100 50

50

AN US

50 0 0

140

120

95

∆𝜔 (Hz)

∆𝜔 (Hz)

200

150

130

165

250

180 170

185

300

190

CR IP T

350

100

150

Ω (rad)

(a) Case 1 400

220 210 200

350

190 180

160 150 140 130 120

200

80

90

100

110

100

Ω (rad)

(b) Case 2

120

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0

ED

∆𝜔 (Hz)

250

50

M

170

300

0 0

200

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 1.5 V

150

100

𝑉𝐷𝐶2 − 𝑉𝐷𝐶1 = 0.5 V

PT

50

CE

0 0

50

100

Ω (rad)

150

200

(c) Case 3

AC

Figure 7: Variations of the differential frequency ∆˜ ω as a function of the base angular velocity Ω in dimensional form for various grain sizes of the naocrystalline silicon and two distinct asymmetric DC voltages with (dashed line) and without (solid line) accounting for the effect of the fringing field. The second natural frequency is considered. Less impact of DC voltage bias on the calibration curves was obtained when operating the microgyroscope around the second natural frequency.

28

150

200