# Effects of Resonator Shape on Nonlinear Resonant Frequencies of a Microresonant Pressure Sensor

Authors: Guang Han; Lizhong Xu
Journal: Shock and Vibration
 (2022)

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## Abstract

A nonlinear dynamics equation for a novel microresonant pressure sensor with a cross-type resonator is proposed. The nonlinear resonant frequencies of the sensor are calculated. Effects of the system parameters and gas pressure on the nonlinear resonant frequencies are investigated. Results show that the effects of nonlinearity on the resonant frequencies increase with increasing length of the resonator, and they decrease with decreasing the clearance between the resonator and the baseplate or gas pressure.


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## Body

## 1. Introduction

Microelectromechanical systems (MEMS), with the advantages of high integration, high precision, low power consumption, and easy mass production, are widely used in intelligent manufacturing, robotics, and other fields [1–4]. MEMS include microsensors, microactuators, and processing circuits [5]. Among them, microsensors have been most widely studied [6].Pressure microsensors are the most mature MEMS devices at present [7]. In the 1960s, the first micro pressure sensor was manufactured using micromachining technology [8]. In the 21st century, the micropressure sensor has developed rapidly. In 2001, Lange used metal oxide semiconductor technology and micromachining technology to produce a microsystem containing sensors with all necessary drives and signal conditioning circuits [9]. Werner and Fahrner used a diamond as a pressing die in a pressure sensor, allowing it to work in a high temperature environment around 300°C [10]. Han et al. proposed a high-precision resonant pressure sensor with two similar resonators; a linear fitting method was used to ensure its output linearity [11]. Li et al. developed a high-sensitivity resonant pressure microsensor, which included a sensing unit made of a silicon-on-insulator (SOI) wafer that was encapsulated by silicon glass, thus improving the conversion efficiency of pressure differences [12]. Shi et al. studied a microresonant pressure sensor with a pair of double-ended tuning forks as resonators [13]. Li et al. proposed a graphene-based resonant pressure sensor, which had the advantages of both a micromechanical sensor and a fiber resonator [14]. Matej et al. studied a ceramic resonant pressure sensor for working at high temperature (up to a maximum temperature of 201°C) [15]. Zhang et al. introduced a micromechanical resonant pressure sensor with two resonators and solved the overload problem [16]. Xiang et al. introduced a resonant pressure microsensor in which the silicon islands are deployed on an SOI wafer to improve the equivalent stiffness and structural stability of the pressure-sensitive diaphragm [17]. Zhao et al. and Yan et al. developed temperature-insensitive silicon resonant pressure sensors with very-low-frequency temperature coefficients [18, 19]. Zamanzadeh et al. proposed a resonant pressure sensor that provided a wider range of tunability and sensing range and a simplified signal-processing circuit [20]. Han et al. developed a new type of resonant pressure sensor based on electrostatic excitation and piezoresistive detection that had a measuring sensitivity of approximately 19 Hz/kPa [21]. Mata-Hernandez et al. introduced a resonant pressure sensor based on CMOS technology and investigated its resonant frequency and quality factor [22]. Alcheikh et al. proposed a kind of resonant pressure sensor with high sensitivity whose sensitivity was significantly improved by switching between its first and third modes [23].As the summary above suggests, numerous studies of microresonant pressure sensors have been completed. Among them, the capacitance-detection technique is easy to implement and is widely used. However, the capacitance between the resonator and the base is very small, so the capacitance change signal in resonance is small, which increases the difficulty of producing the subsequent amplifier circuit. For it, this paper proposes a microresonant pressure sensor with a cross-type resonator to increase the area in the middle of the resonator. It can increase capacitance between the resonator and the base to increase the capacitance change signal in resonance [24].The resonator is an important component of the microresonant pressure sensor. Research on the resonant vibration of the resonator is significant for the design and application of the pressure sensor. Li et al. investigated nonlinear dynamics of a resonant silicon bridge pressure sensor with electrothermal excitation [25]. Zhang et al. studied the resonant frequency of the interfering mode for a resonant pressure sensor and separated it well away from the operational mode [26]. Fu and Xu proposed a multifield coupled dynamics model of a microresonant pressure sensor and studied the influence of various factors on the nonlinear vibration of the sensor [27, 28]. However, for the microresonant pressure sensor with a cross-type resonator, the nonlinear resonant frequencies have not yet been investigated.In this paper, a nonlinear dynamics equation for a microresonant pressure sensor with a cross-type resonator is proposed. The nonlinear resonant frequencies of the microresonant sensor are calculated by using this equation with the multiscale method. Effects of system parameters and gas pressure on the nonlinear resonant frequencies are investigated. Our research results can be used to aid in design of the dynamic performance of microresonant pressure sensors.First, a nonlinear vibration equation of the microresonant pressure sensor is proposed. Second, the vibration equation is solved using the multiscale method, and the nonlinear resonance frequency equation of the sensor is given. Third, changes of the nonlinear resonance frequency of the sensor are analyzed and discussed. At last, a conclusion is given.


## 2. Nonlinear Dynamics Equation of a Pressure Sensor

The coupled dynamics model of the microresonant pressure sensor is shown in Figure1. Here, x is the coordinate of the central axis of the resonator; y is the position coordinate in the direction perpendicular to the x-axis; y (x, t) is the displacement of the resonator in the y-axis direction; L is half the length of the resonator; h is the thickness of the resonator; b is the width of the resonator; Le and be are, respectively, the length and width of the cross structure on the resonator; d0 is the initial clearance between the resonator and the baseplate; q (x, t) is the force per unit length applied to the resonator; U0 is the voltage between the resonator and the baseplate.Figure 1
Coupled dynamics model of the sensor.The dynamics equation of the resonator is [29].(1)EI∂4yx,t∂x4−F∂2yx,t∂x2+ρS∂2yx,t∂t2+Ca∂yx,t∂t=qx,t,where E is the modulus of elasticity; I is the second moment; F is the inner axial force of the resonator; ρ is the density of the resonator material; S is the cross-sectional area of the resonator; Ca is the gas damping factor: Ca=ηb3/d0−y3(η is the dynamic viscosity of the gas).The gas damping force per unit length is [30].(2)fp=Ca∂y∂t.The displacementy (x, t) includes two parts: static displacement y0 and dynamic displacement ∆y. The damping force can be written as a Taylor series at y = y0, which is given as follows:(3)fp=ηb3d0−y03∂∆y∂t+3ηb3d0−y04∆y∂∆y∂t+6ηb3d0−y05∆y2∂∆y∂t+10ηb3d0−y06∆y3∂∆y∂t+⋯.The higher-order terms above the third order are omitted; Eq. (3) can be rewritten as(4)fp=cOp0∂∆y∂t+cOp1∆y∂∆y∂t+cOp2∆y2∂∆y∂t+cOp3∆y3∂∆y∂t,where cOp0=ηb3/d0−y03; cOp1=3ηb3/d0−y04; cOp2=6ηb3/d0−y05; cOp3=10ηb3/d0−y06.Ifε = y0/d0 is defined as the nonlinear parameter, we have(5a)cOp0=ηb3d0−y03=ηb3d02y0y0d01+y0d0+y0d02+3≈εηb3d02y0=εcp0,(5b)cOp1=3ηb3d0−y04=3ηb3d03y0y0d01+y0d0+y0d02+⋯4≈ε3ηb3d03y0=εcp1,(5c)cOp2=6ηb3d0−y05=6ηb3d04y0y0d01+y0d0+y0d02+⋯5≈ε6ηb3d04y0=εcp2,(5d)cOp3=10ηb3d0−y06=10ηb3d05y0y0d01+y0d0+y0d02+⋯6≈ε10ηb3d05y0=εcp3.Let∆q denote the dynamic force per unit length of the resonator along the x-axis; it includes the dynamic electric force ∆q0 and the dynamic van der Waals force ∆qr:(6)∆q=∆q0+∆qr.The Van der Waals force per unit length of the resonator is [31].(7)fv=Hv6πbd0−y3,where HVis the Hamaker constant: HV=10−19J.The Van der Waals force can also be written as a Taylor series aty = y0:(8)fv=Hv6πbd0−y03+Hv2πbd0−y04∆y+Hvπbd0−y05∆y2+5Hv3πbd0−y06∆y3+⋯.Here, the static Van der Waals force is(9)fv0=Hv6πbd0−y03.The higher-order terms above the third order are omitted, and the dynamic Van der Waals force can be expressed as follows:(10)∆qr=kvs1∆y+kvs2∆y2+kvs3∆y3where(11)kvs1=Hv2πbd0−y04=Hv2πbd03y0y0d01+y0d0+y0d02+⋯4≈εHv2πbd03y0kvs2=Hvπbd0−y05=Hvπbd04y0y0d01+y0d0+y0d02+⋯5≈εHvπbd04y0,kvs3=5Hv3πbd0−y06=5Hv3πbd04y02y0d021+y0d0+y0d02+⋯6≈ε5Hv3πbd05y0.The electric force per unit length of the resonator is [32].(12)fe=12εoεrbU2d0−y2.It can also be written as a Taylor series aty = y0:(13)fe=εoεrb2d0−y02U2+εoεrbd0−y03U2∆y+3εoεrb2d0−y04U2∆y2+2εoεrbd0−y05U2∆y3+⋯.Here, the static electric force is(14)fe0=εoεrb2d0−y02U2.The higher-order terms above the third order are omitted, and the dynamic electric force can be written as follows:(15)∆q0=εoεrbd0−y03U2∆y+3εoεrb2d0−y04U2∆y2+2εoεrbd0−y05U2∆y3,where εo is the vacuum permittivity; εr is the relative permittivity;(16)εoεrbd0−y03=εoεrbd02y0y0d01+y0d0+y0d02+⋯3≈εεoεrbd02y0,3εoεrb2d0−y04=3εoεrb2d03y0y0d01+y0d0+y0d02+⋯4≈ε3εoεrb2d03y0,2εoεrbd0−y05=2εoεrbd04y0y0d01+y0d0+y0d02+⋯5≈ε2εoεrbd04y0.Substituting Eqs. (10) and (15) into Eq. (6) yields(17)∆q=εξ1∆y+εξ2∆y2+εξ3∆y3,whereξ1=εoεrbU2/d02y0+Hvb/2πd03y0; ξ2=3εoεrbU2/2d03y0+Hvb/πd04y0; ξ3=2εoεrbU2/d04y0+5Hvb/3πd05y0.Upon substituting Eqs. (4) and (17) into Eq. (1), the coupled dynamics equation of the microresonant pressure sensor is obtained as follows:(18)EI∂4∆y∂x4−F∂2∆y∂x2+ρS∂2∆y∂t2+εcp0+cp1∆y+cp2∆y2+cp3∆y3∂∆y∂t=εξ1∆y+εξ2∆y2+εξ3∆y3.


## 3. Solution of Nonlinear Dynamics Equation

Letting∆y=ϕxqt and substituting it into Eq.(18) yields(19)q″q+εq′ρScp0q+cp1ϕ¯+cp2ϕ¯2q+cp3ϕ¯3q2−εξ2ϕ¯qρS−εξ3ϕ¯2q2ρS=FρSϕ2ϕ−EIρSϕ4ϕ+εξ1ρS.Setting the left and right sides of Eq. (19) to be equal to −ω2 yields(20)ϕ4−FEIϕ2−ω2ρSEI+εξ1EIϕ=0,(21)q″+ω2q+εA1q′+εA2q′q+εA3q′q2+εA4q′q3−εA5q2−εA6q3=0,where A1=cp0/ρS, A2=cp1ϕ¯/ρS, A3=cp2ϕ¯2/ρS, A4=cp3ϕ¯3/ρS, A5=ξ2ϕ¯/ρS, and A6=ξ3ϕ¯2/ρS.(22)ϕ¯=1L∫0Lϕxdx,ϕ¯2=1L∫0Lϕx2dx,ϕ¯3=1L∫0Lϕx3dx.Eq.(20) can be changed into the following form:(23)ϕ4x−α2ϕ2x−β4ϕx=0,where α2=F/EI and β4=ω2ρS/EI+εξ1/EI.The four characteristic roots of Eq.(23) are(24)s1,2=±iλ1=±i−α22+α44+β4,s3,4=±λ2=±α22+α44+β4,where λ1=−α2/2+α4/4+β4 and λ2=α2/2+α4/4+β4.With two local coordinate system set up, the origins are located on the left and right ends of the resonator (see Figure2). In the left coordinate system, the mode function of the left resonator is(25)ϕ1x=C1chλ2x+C2shλ2x+C3cosλ1x+C4sinλ1x.Figure 2
Coordinate systems for the resonator vibration analysis.In the right coordinate system, the mode function of the right resonator is(26)ϕ2x=C5chλ2x+C6shλ2x+C7cosλ1x+C8sinλ1x,where the undetermined constants Cj (j = 1, 2, 3, 4, 5, 6, 7, and 8) and the parameters λj(j = 1, 2) can be determined by boundary conditions and continuity conditions.Since the displacement and rotation angle at the fixed end of the resonator are zero, we have(27)ϕ0=ϕ′0=0.At the center of the resonator, there is a concentrated mass, so the shear force in the resonator is equal to the inertia force of the concentrated mass, i.e.,(28)EIϕ‴L=−12mω2ϕL.When the mode function of the left resonator is antisymmetric to the mode function of the right resonator, it can be concluded that(29)ϕ10=ϕ1L=0,ϕ1′0=0,EIϕ1‴L=−12mω2ϕ1L.When the mode function of the left resonator is symmetric to the mode function of the right resonator, we have, instead,(30)ϕ10=0,ϕ1′0=ϕ1′L=0,EIϕ1‴L=−12mω2ϕ1L.Substituting Eq.(25) into (29) yields(31)C1+C3=0,C2λ2+C4λ1=0,C1chλ2L+C2shλ2L+C3cosλ1L+C4sinλ1L=0,C1EIλ23shλ2L+12mω2chλ2L+C2EIλ23chλ2L+12mω2shλ2L,+C3EIλ13sinλ1L+12mω2cosλ1L+C412mω2sinλ1L−EIλ13cosλ1L=0.This equation has a nonzero solution only if the determinant of the coefficients is equal to zero, i.e.,(32)10100λ20λ1chλ2Lshλ2Lcosλ1Lsinλ1LEIλ23shλ2L+12mω2chλ2LEIλ23chλ2L+12mω2shλ2LEIλ13sinλ1L+12mω2cosλ1L12mω2sinλ1L−EIλ13cosλ1L=0.From this equation, the frequency equation for the antisymmetric mode functions can be written(33)λ13λ2−λ1λ23cosλ1L⋅chλ2L−1+sinλ1L⋅shλ2Lλ14+λ24=0.Substituting Eq.(25) into (30) yields(34)C1+C3=0,C2λ2+C4λ1=0,C1λ2shλ2L+C2λ2chλ2L−C3λ1sinλ1L+C4λ1cosλ1L=0,C1EIλ23shλ2L+12mω2chλ2L+C2EIλ23chλ2L+12mω2shλ2L+C3EIλ13sinλ1L+12mω2cosλ1L+C412mω2sinλ1L−EIλ13cosλ1L=0.This equation also has a nonzero solution only if the determinant of the coefficients is equal to zero, i.e.,(35)10100λ20λ1λ2shλ2Lλ2chλ2L−λ1sinλ1Lλ1cosλ1LEIλ23shλ2L+12mω2chλ2LEIλ23chλ2L+12mω2shλ2LEIλ13sinλ1L+12mω2cosλ1L12mω2sinλ1L−EIλ13cosλ1L=0.It follows from this equation that the frequency equation for symmetric mode functions can be written(36)EIλ14λ2+λ12λ23sinλ1L⋅chλ2L+EIλ13λ22+λ1λ24cosλ1L⋅shλ2L+12mω2λ12−λ22sinλ1L⋅shλ2L+mω2λ1λ2cosλ1L⋅chλ2L−1=0.Eq.(21) can be simplified to(37)q″+ω2q+εA1+εA2q+εA3q2+εA4q3q′−εA5q2−εA6q3=0.Using the multiscale method, we can let(38)q=q0T0,T1,T2+εq1T0,T1,T2+ε2q2T0,T1,T2,(39)ω2=ω021+εσ1+ε2σ2+⋯,where Tn=εntn=0,1,2,…, d/dt=D0+εD1+ε2D2, d2/dt2=D02+2εD0D1+ε2D12+2D0D2.Substituting Eq. (38) into (37) then yields(40)D02+2εD0D1+ε2D12+2D0D2q0+εq1+ε2q2+ω02q0+εq1+ε2q2+εA1+εA2q0+εq1+ε2q2+εA3q0+εq1+ε2q22+εA4q0+εq1+ε2q23D0+εD1+ε2D2q0+εq1+ε2q2−εA5q0+εq1+ε2q22−εA6q0+εq1+ε2q23=0.Setting the sum of the coefficients of the parameterε with the same power equal to zero yields(41a)D02q0+ω02q0=0,(41b)D02q1+ω02q1=−A1D0q0−2D0D1q0+A5q02−A2D0q02+A6q03−A3D0q03−A4D0q04,(41c)D02q2+ω02q2=−A1D1q0−D12q0−2D0D2q0−A2D1q02−A3D1q03−A4D1q04−A1D0q1−2D0D1q1+2A5q0q1−2A2D0q0q1+3A6q02q1−3A3D0q02q1−4A4D0q03q1.Solution of Eq.(41a) can be given as(42)q0=AT1,T2eiω0T0+A¯T1,T2e−iω0T0,where A is an unknown function of a complex variable and A¯is its complex conjugate.Substituting Eq.(42) into Eq. (41b) yields(43)D02q1+ω02q1=−A1Aiω0−2D1Aiω0−3A3A2A¯iω0+3A6A2A¯eiω0T0+A5A2−2A2A2iω0−8A4A3A¯iω0e2iω0T0+A6A3−3A3A3iω0e3iω0T0−4A4A4iω0e4iω0T0+A5AA¯+CC,where CC is the complex conjugate of the previous term.In order to eliminate the secular term in the equation, we let(44)−A1Aiω0−2D1Aiω0−3A3A2A¯iω0+3A6A2A¯=0.From this equation, it follows that(45)D1A=−12A1A−32A3A2A¯+3A6A2A¯2iω0.Substituting Eq.(45) into Eq. (43) yields(46)q1=2A2A2iω0+8A4A3A¯iω0−A5A23ω02e2iω0T0+3A3A3iω0−A6A38ω02e3iω0T04A4A4i15ω0e4iω0T0+A5AA¯ω02+CC.Substituting Eqs. (42) and (46) into Eq. (41c) then yields(47)D02q2+ω02q2=11215A7A42e7iω0T0+245A6A3A4+4iA6A4A65ω0−3iA3A43iω0A3A3−A3A6ω0e6iω0T0+83A5A2A4+8iA5A4A515ω0−15iA2A33iω0A3A3−A3A68ω0+3A2A63iω0A3A3−A3A68ω02−20iA3A42iω0A2A2−A2A5+8iω0A3A4A¯3ω0+16A6A42A¯e5iω0T0+12A4A1A4−iAA23iω0A3A3−A3A6ω0+AA53iω0A3A3−A3A64ω02+4710A5A3A4A¯−iA5A4A6A¯10ω0−6iA2A43iω0A3A3−A3A6A¯ω0−4iA2A32iω0A2A2−A2A5+8iω0A3A4A¯ω0+A2A62iω0A2A2−A2A5+8iω0A3A4A¯ω02e4iω0T0+12A3A1A3−158A4A32A¯+85A4A2A4A¯−352iA4A4A5A¯15ω0−3iA4A3A6A¯ω0+9A4A62A¯8ω02−9iAA33iω0A3A3−A3A6A¯4ω0+3AA63iω0A3A3−A3A6A¯4ω02+485A5A42A¯2−2iAA22iω0A2A2−A2A5+8iω0A3A4A¯ω0+2AA52iω0A2A2−A2A5+8iω0A3A4A¯3ω02−12iA2A4A¯2iω0A2A2−A2A5+8iω0A3A4A¯ω0e3iω0T0+12A2A1A2−52A3A2A3A¯+2A3A1A4A¯−14iA3A3A5A¯ω0−5iA3A2A6A¯2ω0+8A3A5A6A¯ω02−iA23iω0A3A3−A3A6A¯2ω0+A53iω0A3A3−A3A6A¯4ω02−425A4A3A4A¯2−46iA4A4A6A¯25ω0−3iAA43iω0A3A3−A3A6A¯2ω0−4iAA3A¯2iω0A2A2−A2A5+8iω0A3A4A¯ω0+2AA6A¯2iω0A2A2−A2A5+8iω0A3A4A¯ω02e2iω0T0+14A1A2AA¯+34A2A3A2A¯2+34A1A4A2A¯2+3iA2A6A2A¯24ω0+6A5A6A2A¯2ω02+94A3A4A3A¯3+9iA4A6A3A¯34ω0+A6A¯22iω0A2A2−A2A5+8iω0A4A3A¯ω02+CC.In order to eliminate the secular term in this equation, we can set the coefficient of the termeiω0T0 to be zero:(48)240D2Aω3+30A12Aiω2+160A22A2A¯iω2+180A1A3A2A¯iω2+240A2A5A2A¯ω+400A52A2A¯i+405A32A¯2A3iω2+1280A2A4A3A¯2iω2+1600A4A5A3A¯2ω−180A3A6A3A¯2ω+225A62A3A¯2i+2688A42A4A¯3iω2=0.From this equation, it can be concluded that(49)D2A=−iAA128ω0−2iA2A22A¯3ω0−3iA2A1A3A¯4ω0−A2A2A5A¯ω02−5iA2A52A¯3ω03−27iA3A32A¯216ω0−16iA3A2A4A¯23ω0−20A3A4A5A¯23ω02+3A3A3A6A¯24ω02−15iA3A62A¯216ω03−56iA4A42A¯35ω0.Substituting Eq. (49) into Eq. (47) then yields(50)q2=−7A7A4245ω02e7iω0T0−69A6A3A4175ω02+19iA6A4A6175ω03e6iω0T0+−15A5A3264ω02−2A5A2A43ω02−3iA5A4A510ω03−iA5A3A68ω03+A5A6264ω04−26A6A42A¯9ω02e5iω0T0+−11A4A2A315ω02−A4A1A430ω02−19iA4A3A560ω03−iA4A2A65ω03+A4A5A612ω04−547A5A3A4A¯150ω02−139iA5A4A6A¯150ω03e4iω0T0+−A3A222ω02−A3A1A316ω02−5iA3A2A512ω03+A3A5212ω04−39A4A32A¯64ω02−26A4A2A4A¯5ω02+23iA4A4A5A¯30ω03−3iA4A3A6A¯16ω03−3A4A62A¯64ω04−66A5A42A¯25ω02e3iω0T0+−A2A1A26ω02−7A3A2A3A¯3ω02−2A3A1A4A¯3ω02+37iA3A3A5A¯12ω03−2iA3A2A6A¯3ω02−23A3A5A6A¯12ω04−163A4A3A4A¯215ω02−49iA4A4A6A¯215ω03e2iω0T0+5A2A5A6A¯2ω04+AA1A2A¯4ω02+3A2A2A3A¯24ω02+3A2A1A4A¯24ω02+9A3A3A4A¯34ω02+11iA2A2A6A¯24ω03+41iA3A4A6A¯34ω03+CC.Here(51)dAdt=D0A+εD1A+ε2D2A,where D0A=0.Substituting Eqs.(45) and (49) into Eq. (51) yields(52)dAdt=ε−12A1A−32A3A2A¯+3A6A2A¯2iω0+ε2−iAA128ω0−2iA2A22A¯3ω0−3iA2A1A3A¯4ω0−A2A2A5A¯ω02−5iA2A52A¯3ω03−27iA3A32A¯216ω0−16iA3A2A4A¯23ω0−20A3A4A5A¯23ω02+3A3A3A6A¯24ω02−15iA3A62A¯216ω03−56iA4A42A¯35ω0.We then write(53)At=12Kteiθt,where Kt and θt are time-varying functions.Substituting Eq. (53) into Eq. (52) yields(54)K′=−12A1Kε−38A3K3ε−A2A5K3ε24ω02−5A4A5K5ε212ω02+3A3A6K5ε264ω02,(55)θ′=−5A52K2ε212ω03−15A62K4ε2256ω03−3A6K2ε8ω0−A12ε28ω0−A22K2ε26ω0−3A1A3K2ε216ω0−27A32K4ε2256ω0−A2A4K4ε23ω0−7A42K6ε240ω0.Substituting Eqs. (42), (46), and (50) into Eq. (38) yields(56)qt=Kcosφ+ε−A56ω02K2cos2φ−A23ω0K2sin2φ−A43ω0K4sin2φ−A632ω02K3cos3φ−3A332ω0K3sin3φ−A430ω0K4sin4φ+A52ω02K2+ε2−A1A212ω02K2cos2φ−7A2A324ω02K4cos2φ−A1A412ω02K4cos2φ−163A3A4480ω02K6cos2φ−37A3A596ω03K4sin2φ+A2A612ω03K4sin2φ+49A4A6480ω03K6sin2φ−23A5A696ω04K4cos2φ−A228ω02K3cos3φ−A1A364ω02K3cos3φ−39A321024ω02K5cos3φ−13A2A440ω02K5cos3φ−33A42160ω02K7cos3φ+5A2A548ω03K3sin3φ−23A4A5480ω03K5sin3φ+3A3A6256ω03K5sin3φ+A5248ω04K3cos3φ−3A621024ω04K5cos3φ−11A2A3120ω02K4cos4φ−A1A4240ω02K4cos4φ−547A3A44800ω02K6cos4φ+19A3A5480ω03K4sin4φ+A2A640ω03K4sin4φ+139A4A64800ω03K6sin4φ+A5A696ω04K4cos4φ−15A321024ω02K5cos5φ−A2A424ω02K5cos5φ−13A42288ω02K7cos5φ+3A4A5160ω03K5sin5φ+A3A6128ω03K5sin5φ+A621024ω04K5cos5φ−69A3A45600ω02K6cos6φ+19A4A65600ω03K6sin6φ−7A422880ω02K7cos7φ+5A5A68ω04K4+A1A28ω02K2+3A2A332ω02K4+3A1A432ω02K4+9A3A4128ω02K6,where φ=θ+tω0, and K and θ can be determined by Eqs.(54) and (55).Thus, the nonlinear resonant frequency of the sensor can be found from Eq.(39) to be(57)ω=ω01−ε3A6K24−ε2A124+A223+3A1A38+5A526K2+27A32128+2A2A43−3A62128K4+7A4220K6+⋯.The pressure sensor structure is shown in Figure3(a) The pressure membrane used for detecting the pressure is a rectangular membrane fixed at four edges. Here, P denotes the pressure applied to the membrane, and Lf denotes its length and its width. The sizes of the membrane are shown in Figure 3(b).Figure 3
Structure and sizes of the pressure sensor (a) sensor structure (b) sizes of the pressure membrane.
(a)(b)If we letd denote the thickness of the pressure membrane and w denote its elastic displacement, the boundary conditions of the pressure membrane can be written as(58)wx=±Lf=wy=±Lf=0,∂w∂xx=±Lf=∂w∂yy=±Lf=0.The elastic displacement of the pressure membrane satisfying all boundary conditions is then(59)w=A111+cosπxLf1+cosπyLf.Substituting Eq. (59) into the Galerkin equation yields(60)∫∫P−D∇4w1+cosπxLf1+cosπyLfdxdy=0.From Eq. (60), it follows that(61)A11=PLf42π4D,where D=Ed3/121−μ2.The relationship between displacement and stress is known as follows:(62)σx=−Ez1−μ2∂2w∂x2+μ∂2w∂y2.Substituting Eq. (59) into Eq. (62) yields(63)σx=EzA11π21−μ21Lf2cosπxLf1+cosπyLf+μ2Lf2cosπyLf1+cosπxLf.Aty = 0 and z=d/2, the stress on the pressure membrane is(64)σxy=0z=d/2=3Pπ2Lfd22+μcosπxLf+μ.Its average value can be written(65)σ¯x=3Pπ2Lfd2∫0Lf2+μcosπx/Lf+μdxLf=3PLfπd2μ.Thus, the axial forceF in the resonator is(66)F=3PLfπd2μ×h×Lf.When Eq. (66) is substituted into Eq. (18), the effects of the gas pressure on the sensor can be determined.


## 4. Results and Discussion

By using a cross-type resonator, the area of capacitance electrode between the resonator and the substrate is increased, and the capacitance variation is increased under the same resonator vibration amplitude, that is, the strength of the resonant signal of the microresonant pressure sensor is increased.Comparison between the traditional resonator and the cross-type resonator is given in Table1. Here, S is the pole area, ∆y¯ is the average vibration amplitude, ∆C is the capacitance change, and Uout is the output signal voltage. Table 2 shows parameters of the sensor system, and Figures 4 and 5 show effects of nonlinear parameter on resonant frequencies and resonant frequency difference.Table 1
Comparison between traditional and cross-type resonators.
S/m2∆y¯/m∆C/FUout/mVTraditional resonator9.6 × 10−72.0870 × 10−71.4315 × 10−11103.330Cross-type resonator14.4 × 10−72.5584 × 10−72.9446 × 10−11220.064times1.501.232.062.13The pole area of the cross-type resonator is 1.5 times of the traditional resonator. So, the capacitance change and the output signal voltage of the cross-type resonator are about 2 times of the traditional resonator. It is very beneficial to the design and manufacture of the signal-processing circuit.The chosen parameters of the resonator are shown in Table2. Here, the material is silicon (Si), and the nonlinear parameter is ε = 0.25 when other parameters are changed. ε = y0/d0 is the ratio of the static displacement of the resonator to initial clearance. If ε = y0/d0 is above 0.3, the resonator will come into instability state. So, the nonlinear parameter is taken to be 0.25. Using Eqs. (36) and (57), the linear and nonlinear frequencies of the first four orders of resonant vibration of the resonant sensor are obtained, as shown in Table 3.Table 2
Parameters of the sensor system.
L (m)b (m)h (m)m (kg)E (GPa)ρ (kg/m3)4 × 10−35 × 10−41 × 10−53.495 × 10−81902330ε0 (C2·N−1 m−2)εrU0 (V)d0 (m)Hv (J)F (N)8.85 × 10−1211.05 × 10−710 × 10−191Table 3
Comparison of linear and nonlinear resonant frequencies for the first four orders.
Odd mode functionω0 (Hz)ω1 (Hz)η (%)Mode (1, 1)35388.1033054.066.596Mode (2, 2)74560.9172885.342.004Mode (3, 3)114017.01113038.150.859Mode (4, 4)154808.42154115.230.448Even mode functionω0 (Hz)ω1 (Hz)η (%)Mode (1, 1)44117.7542588.683.534Mode (2, 2)80909.6079886.531.265Mode (3, 3)120061.71119279.670.651Mode (4, 4)161300.35160673.290.389Effects of the nonlinear parameter and molecular force on the nonlinear frequencies and the difference between the linear and nonlinear frequencies were investigated (see Figures4–6). Here ω0 is the linear resonant frequency, ω1 is the nonlinear resonant frequency, ∆ωa is the difference between the linear and nonlinear resonant frequencies considering molecular force, ∆ωb is the difference between the linear and nonlinear resonant frequencies without considering molecular force, and η is the relative difference between the linear and nonlinear resonant frequencies. The following can be deduced from Table 3 and Figures 4–6:(1)
The nonlinear resonant frequency is lower than the linear resonant frequency. This shows that nonlinearity has the property of a soft spring. The frequency difference between the linear and nonlinear resonant frequencies decreases gradually from low-order to high-order modes. The frequency difference for the first-order mode is about 13 times that of the fourth-order mode.(2)
The resonant frequency difference of the odd-number order modes is about twice that of the even-number order modes. As the order number increases, the resonant frequency difference for the odd-number modes gradually approaches one for the even-number modes. From Eq.(21), it is known that the nonlinear effects are related to the coefficients A1, A2, A3, A4, A5, and A6, which depend on the vibration modes of the resonator. For the odd-number order modes, these coefficients are larger than those of the even order modes. So, the nonlinearity has more obvious effects on the odd-number order modes than those on the even-number order modes.(3)
With an increase of the nonlinear parameter ɛ, the nonlinear resonant frequencies decrease. When the nonlinear parameter ɛ increases from 0.1 to 0.25, the resonant frequency of the odd-number order mode decreases by 5.3%, and the resonant frequency of the even-number order mode decreases by 2.9%. The nonlinear parameter has a more obvious effect on the resonant frequencies for the odd-number order modes than that for the even-number order modes. With an increase of the nonlinear parameter ɛ, the frequency difference between the linear and nonlinear resonant frequencies increases gradually as well. For odd-number order modes, the frequency difference increases more significantly than for the even-number order modes.(4)
The difference between the linear and nonlinear resonant frequencies is larger when molecular force is considered than when it is not considered. The frequency difference for the odd-number order modes is larger than for the even-number order modes. As the length of the resonator increases, the frequency difference between the linear and nonlinear resonant frequencies increases gradually. This means that the effect of the molecular force on the resonant frequencies increases along with the resonator length.Figure 4
Effects of the nonlinear parameter on resonant frequencies. (a) Odd. (b) Even.
(a)(b)Figure 5
Effects of the nonlinear parameter on resonant frequency difference. (a) Odd. (b) Even.
(a)(b)Figure 6
Effects of molecule force and resonator length on resonant frequency difference. (a) Odd. (b) Even.
(a)(b)By changing the length and thickness of the resonator and the clearance between the resonator and the base plane, the variation of resonant frequencies with and without considering nonlinearity can be further studied. The results are shown in Figures7–10.(1)
With an increase in the length of the resonator, its resonant frequency decreases, and the decrease of high-order resonant frequencies is more evident than those of low order. When the length of the resonator increases from 2.5 × 10−3 to 4 × 10−3 m, the resonant frequency of the first odd-number order decreases by 16.6%, while the resonant frequency of the fourth odd-number order decreases by 39.3%. At the same time, the resonant frequency of the first even-number order decreases by 19.8%, while the resonant frequency of the fourth even-number order decreases by 42.2%.With an increase in the length of the resonator, the frequency difference between the linear and nonlinear resonant frequencies increases. This means that effects of the nonlinearity on the resonant frequencies increase as the resonator length increases. For the odd-number modes, the effects of the nonlinearity are more significant than those for the even-number modes.(2)
With an increase in the clearance between the resonator and the base plane, the resonant frequency increases, and the increases of the low-order resonant frequencies are more obvious than those of the high-order modes. When the clearance increases from 3 × 10−7 m to 4.5 × 10−7 m, the first odd-number order resonant frequency increases by 6.53% and the fourth odd-number order resonant frequency increases by 0.35%. At the same time, the first even-number order resonant frequency increases by 3.50% and the fourth even-number order resonant frequency increases by 0.29%. With an increase of the clearance between the resonator and the base plane, the frequency difference between the linear and nonlinear resonant frequencies decreases. This means that effects of the nonlinearity on the resonant frequencies decreases as the clearance between the resonator and the base plane decreases.(3)
With an increase in the thickness of the resonator, its resonant frequency decreases, and the decrease of the high-order resonant frequency is more significant than that of the low-order modes. As the thickness of the resonator increases from 2 × 10−6 m to 10 × 10−6 m, the resonant frequency of the first-order mode decreases by 48.5%, while the resonant frequency of the fourth-order mode decreases by only about 1/4 as much. With an increase of the thickness of the resonator, the frequency difference between the linear and nonlinear resonant frequencies decreases. This means that effects of the nonlinearity on the resonant frequencies decrease as the resonator thickness decreases.Figure 7
Changes of the resonant frequencies with system parameters (odd modes) (a) length changes (b) clearance changes (c) thickness changes.
(a)(b)(c)Figure 8
Changes of the resonant frequency differences with system parameters (odd modes) (a) length changes (b) clearance changes (c) thickness changes.
(a)(b)(c)Figure 9
Changes of the resonant frequencies with system parameters (even modes) (a) length changes (b) clearance changes (c) thickness changes.
(a)(b)(c)Figure 10
Changes of the resonant frequency differences with system parameters (even modes). From these figures, the following conclusions can be reached: (a) length changes (b) clearance changes (c) thickness changes.
(a)(b)(c)In brief: the effects of the nonlinearity on the resonant frequencies increase for larger length, smaller clearance, and smaller thickness.Effects of the pressure on the resonant frequencies for the sensor have been investigated and are shown in Figures11 and 12. Here are conclusions that can be drawn:Figure 11
Changes of the resonant frequencies with pressure (a) odd modes (b) even modes.
(a)(b)Figure 12
Changes of the resonant frequency differences with pressure (a) odd modes (b) even modes.
(a)(b)With an increase of gas pressure, the resonant frequencies increase, and the increase of the high-order resonant frequencies is greater than that of the low-order modes. When the gas pressure increases from 0 to 3 × 105 Pa, the resonant frequency of the first odd-number order increases by 77.8% and the resonant frequency of the fourth odd-number order decreases by 116.7%. At the same time, the resonant frequency of the first even-number order increases by 78.95%, and the resonant frequency of the fourth even-number order decreases by 142.11%. As the gas pressure increases, the frequency difference between the linear and nonlinear resonant frequencies decreases. For the low-order modes, effects of the gas pressure on the resonant frequency and the resonant-frequency differences are more significant than for the high-order modes. This means that effects of the nonlinearity on the resonant frequencies decrease as the pressure decreases. For the even-number modes, the effects of the pressure are more significant than for the odd-number modes.In brief: with increasing gas pressure, the resonant frequency of the resonator increases, effects of the nonlinearity on the resonant frequency weaken, and effects of the gas pressure on the resonant frequency are more significant for the even-number modes than for the odd-number modes.Using software MATLAB, Dsolve function is used to resolve the differential equation (37), and the nonlinear resonant frequencies are obtained. The results are shown in Table 4. Here, ω2 is the nonlinear resonant frequencies obtained by Dsolve function, and η2 is the relative error between the numerical simulations and the multiscale method. Results show that the maximum relative error between the nonlinear resonant frequencies given by the numerical simulations and the multiscale method is 3.528%. It illustrates the multiscale method in the research.Table 4
Comparison of the multiscale method and the numerical method.
Odd mode functionω1 (Hz)ω2 (Hz)η2 (%)Mode (1, 1)33054.0633988.172.826Mode (2, 2)72885.3474654.992.428Mode (3, 3)113038.15115868.632.504Mode (4, 4)154115.23157837.112.415Even mode functionω1 (Hz)ω2 (Hz)η2 (%)Mode (1, 1)42588.6843686.382.577Mode (2, 2)79886.5382704.933.528Mode (3, 3)119279.67122441.772.651Mode (4, 4)160673.29164764.032.546


## 5. Conclusions

In this paper, a nonlinear dynamics equation for a novel microresonant pressure sensor is proposed. Effects of the system parameters and gas pressure on the nonlinear resonant frequencies are investigated. Results show the following conclusions:(1)
The nonlinear resonant frequency is less than the linear resonant frequency. The frequency difference between the linear and nonlinear resonant frequencies decreases gradually from low-order to high-order modes.(2)
The frequency difference between the linear and nonlinear resonant frequencies is larger when molecular force is considered than when it is not considered.(3)
The effects of the nonlinearity on the resonant frequencies increase with increasing resonator length or decreasing the clearance between the resonator and the baseplate. With increasing gas pressure, the resonant frequency of the resonator increases.



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