The micro resonant pressure sensor outputs the frequency signals where the distortion does not take place in a long distance transmission. As the dimensions of the sensor decrease, the effects of the van der Waals forces should be considered. Here, a coupled dynamic model of the micro resonant pressure sensor is proposed and its coupled dynamic equation is given in which the van der Waals force is considered. By the equation, the effects of the van der Waals force on the natural frequencies and vibration amplitudes of the micro resonant pressure sensor are investigated. Results show that the natural frequency and the vibrating amplitudes of the micro resonant pressure sensor are affected significantly by van der Waals force for a small clearance between the film and the base plate, a small initial tension stress of the film, and some other conditions.
1. Introduction
Microelectromechanical Systems (MEMS) have advantages such as compact structure, low cost, small power loss, high response speed, and high accuracy [1, 2]. The micro resonant sensor is more attractive because it outputs the frequency signals where distortion error does not occur and it is suitable for the distant range transmission.
In 1990-1991, the micro resonant pressure sensors with heat excitation and electromagnetic excitation were developed [3, 4]. In this year, a micro resonant pressure sensor with electrostatic excitation was proposed which can achieve an accuracy of 100 ppm at the temperature range of the automobile operation [5]. A new alloy based on transition metals was developed to obtain large amplitude of the resonant longitudinal magnetoelastic waves which can be used to improve performance of the micro resonant sensor [6]. A type of NEMS double Si3N4 resonant beams pressure sensor was presented and the sensitivity of the sensor is getting to 498.24 Hz/kPa [7]. An electrothermally excited dual beams silicon resonant pressure sensor with temperature compensation was proposed and the experimental results indicate that the maxim residual error is 1.8 kPa in the working temperature range from −40 to 60°C [8]. The nonlinear dynamics of a resonant silicon bridge pressure sensor with electrothermal excitation was investigated in which the measured pressure, the heating effect of the electrothermal excitation, and the residual internal force in the bridge were considered [9]. The sensitivity of the micro resonant pressure sensor with docks was investigated [10]. A novel resonant pressure sensor with an improved micromechanical double-ended tuning fork resonator packaged in dry air at atmospheric pressure was presented in which the fundamental frequency of the resonant pressure sensor is approximately 34.55 kHz with a pressure sensitivity of 20.77 Hz/kPa [11]. A new stress isolation method based resonant pressure sensor was presented to minimize thermal stresses arising from device packaging due to thermal mismatches between the silicon sensor body and its housing materials [12].
In a word, a number of studies about micro resonant pressure sensor have been done. However, in a micro resonant pressure sensor, as the clearance between the resonant film and basement is small enough, the effects of the van der Waals force will become obvious.
The influence of surface effects on the pull-in instability of a cantilever nanoactuator was investigated incorporating the influence of the Casimir attraction and van der Waals force [13, 14]. Besides it, the effects of the molecular forces on the free vibration of electromechanical integrated electrostatic harmonic actuator were studied as well [15].
Above-mentioned studies mainly focus on dynamics performance of the micro ring (see [15]) and micro beam (other references) which is only controlled by one partial differential equation. In our micro resonant pressure sensor, the resonator is the micro film in which the dynamics performance is controlled by two partial differential equations. So, the effect problem of the van der Waals force on the dynamics performance of a micro resonant pressure sensor is more complicated and has not been resolved yet.
In this paper, a coupled dynamic model of the micro resonant pressure sensor is proposed and its coupled dynamic equation is given in which the van der Waals force is considered. Using these equations, the effects of the van der Waals force on the natural frequencies and vibration amplitudes of the micro resonant pressure sensor are investigated. Results show that the natural frequency and the vibrating amplitudes of the micro resonant pressure sensor are affected significantly by van der Waals force for a small clearance between the film and the base plate, a small initial tension stress of the film, and some other conditions. These results can be used to improve design about dynamics performance for the micro resonant pressure sensor.
2. Vibration Equations
Figure 1 illustrates a micro film in the micro resonant pressure sensor. The electrostatic force and the van der Waals force are applied to the micro film. Its boundary condition is that two sides are fixed and two sides are free. Here, the dynamics partial differential equation of the film under uniform tension is not applicable. Two dynamics partial differential equations of the orthotropic film can be used. The vibration equations of the orthotropic micro film are [16](1a)ρh∂2w∂t2-σ0x+∂2φ∂y2h∂2w∂x2-σy0+∂2φ∂x2h∂2w∂y2=q,(1b)1E1∂4φ∂y4+1E2∂4φ∂x4=∂2w∂x∂y2-∂2w∂x2∂2w∂y2,where ρ is the mass density of the film, t is the time, v is the initial clearance between the micro film and base plate, U is the voltage between the micro film and base plate, w is the transverse displacement of the film, h is the thickness of the film, φ is the stress function, σ0x is the initial tension stress in x direction of the film, σy0 is the initial tension stress in y direction of the film, x is the coordinate in the film length direction, y is the coordinate in the film width direction, E is the modulus of elasticity of the micro film material, and q(x,y,t) is the transverse load per unit area on the film.
Multifields coupled dynamic model of a micro film.
The electrostatic force per unit area is(2)qe=FeA=ε0εr2Uv-w2,where Fe is the electrostatic force applied to the micro film, A is the area of the micro film, ε0 is permittivity constant of free space, and εr is relative dielectric constant of the insulating layer.
The van der Waals force per unit area between the film and fixed plate is(3)qr3=A6πv-w3,where A is the Hamaker constant: A=10-19 J.
The damping force per unit area from air is(4)qc=ca∂w∂t.Here, ca is the damping coefficient of the gas.
Thus, the total force per unit area on the film is(5)qx,y,t=qe+qr3+qc.
The displacement w of the micro film consists of a static component ws and a dynamic one Δw: (6)w=ws+Δw.
The load q(x,y,t) of static (qs) and dynamic (Δq) components is(7)qx,y,t=qs+Δq,(8)qs=U02ε0εr2v-ws2+A6πv-ws3.
From Δq=(dq/dw)Δw, we know that(9)Δq=U02ε0εrv-ws3Δw+A2πv-ws4Δw-ca∂Δw∂t,where ca=μb3/v-ws3 and μ is the gas viscosity: μ=1.86×10-5 N·S·m−2.
Substituting (6) and (7) into (1a) and (1b) yields the following equations:(10a)σ0x+∂2φ∂y2h∂2ws∂x2-σy0+∂2φ∂x2h∂2ws∂y2=qs,(10b)1E1∂4φ∂y4+1E2∂4φ∂x4=∂2ws∂x∂y2-∂2ws∂x2∂2ws∂y2,(11a)ρh∂2Δw∂t2-σ0x+∂2Δφ∂y2h∂2Δw∂x2-σy0+∂2Δφ∂x2h∂2Δw∂y2=Δq,(11b)1E1∂4Δφ∂y4+1E2∂4Δφ∂x4=∂2Δw∂x∂y2-∂2Δw∂x2∂2Δw∂y2.
3. Free Vibration
For a micro pressure sensor, two ends of the film are fixed and other two ends of the film are free; the boundary conditions are(12)Δw0,y,t=Δwa,y,t=0,∂Δw∂yy=0=∂Δw∂yy=b=0.
For the boundary conditions, the solutions of (11a) and (11b) can be given as(13a)Δwx,y,t=Wx,yTt,(13b)Δφx,y,t=ϕx,yT2t.Here, (13c)Wx,y=Asinmπxacosnπyb.
Thus(13d)Δwx,y,t=AsinmπxacosnπybTt.
Substituting (13b) and (13d) into (11b) yields(14)1E1∂4ϕ∂y4+1E2∂4ϕ∂x4=m2n2π42a2b2cos2mπxa-cos2nπyb.
Letting ϕx,y=αcos(2mπx/a)-βcos(2nπy/b), and substituting it into (14), yields(15)α=E2n2a232m2b2,β=E1m2b232n2a2.
Substituting it into (13b) yields(16)Δφx,y,t=E2n2a232m2b2cos2mπxa-E1m2b232n2a2cos2nπybT2t.
Letting E=U02ε0ε/v-w03+A/2πv-w04 (here, w0 is the average static displacement of the micro film) and substituting (13b), (13d), and (9) into (11a) and using Galerkin method yield(17)T¨+π2ρm2a2σ0x+n2b2σy0-EρhT+CaρhT˙+3π416ρE1m4a4-E2n4b4T3=0.Letting ε=h2/ab, (17) can be changed into the following form:(18)T¨+ω02T=εα1T3+α2T˙,where ω02=(π2/ρ)(m2/a2)σ0x+(n2/b2)σy0-E/ρh, α1=-(3π4ab/16ρh)E1m4/a4-E2n4/b4, and α2=-abCa/ρh2.
Letting fT,T˙=α1T3+α2T˙ and ψ=ω0t+θ, if ε=0, then (18) can be changed into the following form:(19)dTdt=-aω0sinψ.Letting T=acosψ, here a and ψ are the function of the time. Thus(20)dTdt=dadtcosψ-aω0+dθdtsinψ.From (19) and (20), we can give(21)dadtcosψ-adθdtsinψ=0,(22)d2Tdt2=-dadtω0sinψ-aω0ω0+dθdtcosψ.
Substituting T=acosψ and (22) into (18) yields(23)-dadtω0sinψ-aω0dθdtcosψ=εfacosψ,-aω0sinψ.From (21) and (23), we can give(24a)dadt=-εω0facosψ,-aω0sinψsinψ,(24b)dθdt=-εaω0facosψ,-aω0sinψcosψ.The right parts of (24a) and (24b) can be written in Fourier series:(25a)facosψ,-aω0sinψsinψ=A0a+∑n=1∞Ancosnψ+Bnsinnψ,(25b)facosψ,-aω0sinψcosψ=C0a+∑n=1∞Cncosnψ+Dnsinnψ,where(26)A0a=12π∫02πsinψfacosψ,-aω0sinψdψ=-12α2aω0,C0a=12π∫02πcosψfacosψ,-aω0sinψdψ=38α1a3.
In (25a) and (25b), only the average values A0a and C0a are kept. Substituting (25a) and (25b) into (24a) and (24b) yields(27a)dadt=-εω0A0a=12εα2a,(27b)dθdt=-εaω0C0a=-38εα1a2ω0,(27c)dψdt=ω0-εaω0C0a=ω0-38εα1a2ω0.
From (27a), (27b), and (27c), we know that(28a)a=a0eα2εt/2,(28b)ψ=ω0-3α1a2ε8ω0t+ψ0,where a0 is the initial displacement and ψ0 is the initial phase. Letting ψ0=0, we obtain(29a)Tt=a0eα2εt/2cosω0-3α1a02eα2εtε8ω0t,(29b)ω=ω0-3α1a02eα2εtε8ω0.
Substituting (29a) and (29b) into (13d) yields(30)wx,y,t=∑m=1∞∑n=1∞sinmπxacosnπyba0eα2εt/2cosω0-3α1a02eα2εtε8ω0t.
4. Results and Discussions
Above equations are utilized for the free vibration analysis of the micro resonant pressure sensor. The parameters of the numerical example are shown in Table 1 (here, a0=2×10-7m, U0=0.4V, and σy0=0). Natural frequencies of the micro resonant pressure sensor under various clearances are shown in Table 2 (here, σ0x=5.0×103KN/m). Natural frequencies of the micro resonant pressure sensor under various initial tension stresses are shown in Table 3 (here, v=5.0×10-7 m). ω is the natural frequency of the sensor without considering the van der Waals force, ωf is the natural frequency of the sensor with considering the van der Waals force, and η is the relative error between them. From Tables 2 and 3, the following observations are worth noting.
Parameters of the micro film.
a
b
h
ε0
v
E
ρ
(mm)
(mm)
(μm)
(C2⋅N−1⋅m−2)
(μm)
(GPa)
(kg/m3)
2
1
5
8.85 × 10−12
0.5
190
2330
Natural frequency of the micro pressure sensor for various clearances (rad/s).
ω
ωf
η (%)
ω
ωf
η (%)
v=5.0×10-7 m
Mode (1,1)
28163.51
24990.48
11.27
Mode (3,3)
207730.60
207324.24
0.196
Mode (2,2)
129142.32
128487.67
0.507
Mode (4,4)
283224.23
282926.33
0.105
v=5.5×10-7 m
Mode (1,1)
49598.20
48671.80
1.868
Mode (3,3)
211704.51
211489.39
0.102
Mode (2,2)
135442.01
135105.52
0.248
Mode (4,4)
286151.65
285992.53
0.056
v=6.0×10-7 m
Mode (1,1)
58272.73
57813.02
0.789
Mode (3,3)
213903.08
213728.30
0.082
Mode (2,2)
138853.40
138661.11
0.138
Mode (4,4)
287782.02
287689.29
0.032
Natural frequency of the micro pressure sensor for various initial tension stresses (rad/s).
ω
ωf
η (%)
ω
ωf
η (%)
σ0x=4.5×103 KN/m
Mode (1,1)
16238.77
9748.79
39.966
Mode (3,3)
195925.07
195494.18
0.220
Mode (2,2)
120663.99
119963.07
0.581
Mode (4,4)
267851.08
267536.05
0.118
σ0x=5.0×103 KN/m
Mode (1,1)
28163.51
24990.48
11.27
Mode (3,3)
207730.60
207324.24
0.196
Mode (2,2)
129142.32
128487.67
0.507
Mode (4,4)
283224.23
282926.33
0.105
σ0x=5.5×103 KN/m
Mode (1,1)
36368.50
33970.71
6.594
Mode (3,3)
218900.35
218514.77
0.176
Mode (2,2)
137097.33
136480.84
0.450
Mode (4,4)
297804.85
297521.54
0.095
(1) Without considering the van der Waals force, the natural frequencies of the micro sensor are larger than those in the case of considering the van der Waals force. It is because the van der Waals force system is equivalent to a soft spring system. The van der Waals force can cause decrease of the natural frequencies of the micro sensor.
(2) The deviation between the natural frequencies of the micro sensor with and without the van der Waals force decreases with increasing the order number of the mode. For mode (1,1), the relative error between the natural frequencies is 11.27% (v=5.0×10-7 m). For mode (4,4), the relative error between the natural frequencies is 0.1% (v=5.0×10-7 m). It shows that influence of the van der Waals force on the natural frequencies decreases with increasing the order number of the mode.
(3) At an initial tension stress, the natural frequency of the sensor increases significantly with increasing the clearance between the film and the base plate.
If the clearance grows, the deviation between the natural frequencies of the sensor with and without the van der Waals force drops. At v=5.0×10-7 m, the relative error between the natural frequencies with and without the van der Waals force is 11.27% for mode (1,1). At v=6.0×10-7 m, the relative error between the natural frequencies with and without the van der Waals force is 0.79%.
(4) At a constant clearance between the film and the base plate, the natural frequency of the sensor increases significantly with increasing the film tension. As the film tension grows, the deviation between the natural frequencies of the sensor with and without the van der Waals force decreases. At σ0x=4.5×103KN/m, the relative error between the natural frequencies with and without the van der Waals force is 39.97% for mode (1,1). At F=σ0x=5.5×103KN/m, the relative error between the natural frequencies with and without the van der Waals force is 6.59% for mode (1,1).
Hence, the effects of the van der Waals force on the natural frequency of the micro resonant pressure sensor should be considered for a small clearance between the film and the base plate, a small initial tension stress of the film, and a low order mode of the vibrations.
The effects of the van der Waals force on the vibrating amplitudes of the film center are investigated for mode (1,1) and various system parameters (see Figures 2–6). They show the following.
Effects of the van der Waals force on vibrating amplitude of the film center for various a.
With van der Waals force
Without van der Waals force
Difference of the vibrating amplitudes
Effects of the van der Waals force on vibrating amplitude of the film center for various b.
With van der Waals force
Without van der Waals force
Difference of the vibrating amplitudes
Effects of the van der Waals force on vibrating amplitude of the film center for various h.
With van der Waals force
Without van der Waals force
Difference of the vibrating amplitudes
Effects of the van der Waals force on vibrating amplitude of the film center for various σ0x.
With van der Waals force
Without van der Waals force
Difference of the vibrating amplitudes
Effects of the van der Waals force on vibrating amplitude of the film center for various U0.
With van der Waals force
Without van der Waals force
Difference of the vibrating amplitudes
(1) Due to the effects of the damping, the free vibration of the film center is periodic vibration with the amplitude decay. As the length a of the micro film grows, the vibrating amplitudes of the film center drop more rapidly with the time. When the van der Waals force is not considered, the vibrating amplitudes of the film center drop more rapidly with time compared to the case when the van der Waals force is considered. As the length a of the micro film grows, the effects of the van der Waals force on the vibrating amplitudes of the film center become large.
(2) As the width b of the micro film grows, the vibrating amplitudes of the film center drop. However, the amplitude decay does not occur. When the van der Waals force is not considered, the vibrating amplitudes of the film center are larger than those in the case when the van der Waals force is considered. As the width b of the micro film grows, the effects of the van der Waals force on the vibrating amplitudes of the film center become small.
(3) As the thickness h of the micro film grows, the vibrating amplitudes of the film center drop more slowly with time. When the van der Waals force is not considered, the vibrating amplitudes of the film center drop first more rapidly and then more slowly with time compared to those in the case when the van der Waals force is considered. As the thickness h of the micro film grows, the effects of the van der Waals force on the vibrating amplitudes of the film center first become small and then become large.
(4) As the initial tension stress of the micro film grows, the vibrating amplitudes of the film center drop more slowly with time. When the van der Waals force is not considered, the vibrating amplitudes of the film center drop first more rapidly and then more slowly with time compared to those in the case when the van der Waals force is considered. As the initial tension stress of the micro film grows, the effects of the van der Waals force on the vibrating amplitudes of the film center first become small and then become large.
(5) As the voltage between the micro film and the back plate grows, the vibrating amplitudes of the film center drop more rapidly with time. When the van der Waals force is not considered, the vibrating amplitudes of the film center drop more rapidly with time compared to those in the case when the van der Waals force is considered. As the voltage grows, the effects of the van der Waals force on the vibrating amplitudes of the film center become large.
In a word, the effects of the van der Waals force on the vibrating amplitudes of the film become large under some conditions which include a large voltage between the micro film and the back plate, a large film length and thickness, a small film width, and a small initial tension stress of the micro film.
5. Conclusions
In this paper, a coupled dynamic model of the micro resonant pressure sensor is proposed and its coupled dynamic equation is given in which the van der Waals force is considered. By the equation, the effects of the van der Waals force on the natural frequencies and vibration amplitudes of the micro resonant pressure sensor are investigated. Results show the following.
(1) The effects of the van der Waals force on the natural frequency of the micro resonant pressure sensor should be considered for a small clearance between the film and the base plate, a small initial tension stress of the film, and a low order mode of the vibrations.
(2) The effects of the van der Waals force on the vibrating amplitudes of the film should be considered for a large voltage between the micro film and the back plate, a large film length and thickness, a small film width, and a small initial tension stress of the micro film.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This project is supported by Key Basic Research Foundation in Hebei Province of China (13961701D).
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