Analytical modeling of squeeze air film damping of biomimetic MEMS directional microphone

Analytical modeling of squeeze air film damping of biomimetic MEMS directional microphone

Journal of Sound and Vibration 375 (2016) 422–435 Contents lists available at ScienceDirect Journal of Sound and Vibration journal homepage: www.els...

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Journal of Sound and Vibration 375 (2016) 422–435

Contents lists available at ScienceDirect

Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Analytical modeling of squeeze air film damping of biomimetic MEMS directional microphone Asif Ishfaque, Byungki Kim n School of Mechatronics Engineering, Korea University of Technology and Education, 1600 Chungjeol-ro, Byeongchun-myeon, Cheonan, Chungnam, 31253, Republic of Korea

a r t i c l e in f o

abstract

Article history: Received 1 September 2015 Received in revised form 24 March 2016 Accepted 22 April 2016 Handling Editor D.J Wagg Available online 4 May 2016

Squeeze air film damping is introduced in microelectromechanical systems due to the motion of the fluid between two closely spaced oscillating micro-structures. The literature is abundant with different analytical models to address the squeeze air film damping effects, however, there is a lack of work in modeling the practical sensors like directional microphones. Here, we derive an analytical model of squeeze air film damping of first two fundamental vibration modes, namely, rocking and bending modes, of a directional microphone inspired from the fly Ormia ochracea's ear anatomy. A modified Reynolds equation that includes compressibility and rarefaction effects is used in the analysis. Pressure distribution under the vibrating diaphragm is derived by using Green's function. From mathematical modeling of the fly's inspired mechanical model, we infer that bringing the damping ratios of both modes in the critical damping range enhance the directional sensitivity cues. The microphone parameters are varied in derived damping formulas to bring the damping ratios in the vicinity of critical damping, and to show the usefulness of the analytical model in tuning the damping ratios of both modes. The accuracy of analytical damping results are also verified by finite element method (FEM) using ANSYS. The FEM results are in full compliance with the analytical results. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Directional microphone Squeeze air film damping Modified Reynolds equation Green's function Finite element method

1. Introduction Directional microphones are the talk of the town since last two decades because of their ability to enhance the quality of sound, and to facilitate the hearing aid user's to better understand the speech in noisy environments [1]. In 1995, Miles et al. boosted this work by discovering the remarkable property of the fly Ormia ochracea in enhancing the direction sensitivity cues [2]. They designed a capacitive type rectangular shaped microphone, which was suspended from two opposite sides with the help of two hinged torsional bars [3]. Later, the same group designed an optical readout scheme on the directional microphone. Their scheme worked on the analysis of the diffracted laser beam from the optical gratings, which were designed on both sides of the microphone [4,5]. Hall et al. extended this work by designing a piezoelectric relative displacement measuring scheme. They designed four piezoelectric materials contained cantilever springs on four sides of the microphone. The displacement of the diaphragm induced strain in the attached springs, which was converted into voltages by piezoelectric effect [6,7]. The above-mentioned groups had designed the directional microphones by providing a clear

n

Corresponding author. E-mail addresses: [email protected] (A. Ishfaque), [email protected] (B. Kim).

http://dx.doi.org/10.1016/j.jsv.2016.04.031 0022-460X/& 2016 Elsevier Ltd. All rights reserved.

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path for the air to leave the system. They kept the distance between the moving diaphragm and the substrate in the range of 400–500 μm, so the squeeze air film damping effects were negligible in the dynamic response of the microphone. The concept of taking advantage of squeeze air film damping to enhance the directional sensitivity cues by designing a perforated diaphragm was first introduced by Kim et al. [8]. The squeezed air film between moving and fixed surfaces induces significant damping effects in microelectromechanical systems (MEMS) due to their miniature dimensions. The main method to study these damping effects is to introduce holes in the structure. The careful modeling of different hole sizes can bring the response according to the user's intention. Different analytical models have been proposed in the literature to address the squeeze air film damping effects. The basic equation used for the analysis is Reynolds equation. Bao et al. derived a modified Reynolds equation by adding the air penetration terms, and solved for a finite dimensions thick hole plate [9]. In 2007, compressibility effect and rarefaction effect were introduced in Bao's model by Pandey et al. [10]. The same group performed the squeeze air film damping analysis on a rectangular plate having transverse motion. First, they found the pressure under the whole plate, subtracted the hole pressure from the total pressure, then separately calculated the damping effects through holes and added them in the final results. They validated their results by comparing them with other renowned approaches [11]. The squeeze air film damping analysis of a perforated circular plate having transverse motion was studied by Li et al. [12]. They also extended their work for the torsional motion of a rectangular perforated plate [13]. All these authors worked on a one generalized equation for the whole system, including all factors. Mohite et al. and Homentcovschi et al. adopted another approach for the squeeze air film damping analysis. They divided the whole plate into uniformly distributed cells, and each cell had one hole. They solved the model for one cell and then multiplied the derived damping results with the remaining total cells [14–16]. Nowadays, numerical techniques, like FEM, are the most accurate one to predict the squeeze air film damping response of miniature devices. On the other hand, these techniques are very time consuming and non-transparent [17]. These techniques are also lacking in giving clear idea to the user about the impact of individual parameters changes on the final results. Thus, there is a need to perform an analytical modeling of the structure to get a better insight of the dynamics of the structure. In this paper, we present an analytical model of squeeze air film damping of a perforated biomimetic MEMS directional microphone. The microphone design is novel because it takes advantage of squeeze air film damping to enhance the direction sensitivity cues. The analytical model is derived for both rocking and bending modes of vibration of the microphone. Furthermore, Green's function is used for first time to extract the damping formulas of both modes of vibration. The flow of the paper is as, in Section 2, we present our unique perforated model of the directional microphone, followed by the mathematical modeling to observe the effect of damping ratios on the dynamics of the structure. Section 3 deals with the analytical modeling of squeeze air film damping of the rocking mode of the diaphragm. Section 4 deals with the analytical modeling of squeeze air film damping of the bending mode of the diaphragm. In both modes, the modified Reynolds equation from Pandey et al. [10] is used for modeling. The formulas for damping and spring constants are derived by using Green's function. In Section 5, we vary different microphone parameters to bring the damping ratio of the microphone near to the critical damping. Validation of the analytical results with Li et al. [13] and ANSYS results are also discussed in this section. Finally, the conclusion is presented in Section 6.

2. Biomimetic directional microphone A merely 500 μm separated ears structure of the fly Ormia ochracea is capable of enhancing the direction sensitivity cues. The dynamic model of ears is such that it forms two fundamental modes of vibration in response to incoming sound. The first rocking mode is formed by the oscillations of the tympana at 180° out of phase. The second bending mode is formed by the in-phase oscillations of the tympana [18,19]. These modes combine together and help the fly to localize the sound source. Here, we mimic the ears structure of the fly Ormia ochracea and design a mechanical model. The model is designed in such a way that it takes advantages of the squeeze air film damping effects to enhance the directional sensitivity cues by careful modeling of the hole's ratio and the initial gap thickness between the fixed substrate and the moving diaphragm. The designed rectangular model with circular holes and two torsional beams hinged on opposite fixed supports is shown in Fig. 1. The model has two inertia (I) as to mimic the left and right sides of the ears tympana. The optical gratings are included

Fig. 1. Directional microphone design.

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in the microphone design to measure the relative displacement. The K 1 and K 2 represent the stiffness constants of left and right sides respectively. The C 1 and C 2 represent the damping constants of left and right sides respectively. The K t represents the stiffness constant of the torsional beams. The T 1 and T 2 are torques caused by outside incoming sound pressure. For small angular rotation θ1 and θ2 , the governing equation of the system can be written as #" # " # #" # "  " # " C1L K1 þ Kt T1 0 Kt θ1 θ_ 1 I 0 θ€ 1 þ þ ¼ (1) Kt K2 þ Kt θ2 T2 0 C2L θ_ 2 θ€ 2 0 I The significance of the critical damping ratio in enhancing the directional sensitivity cues of the fly's inspired directional microphone was first mentioned by Miles et al. [2]. To verify this concept, we test the model on different values of damping ratios. The values for stiffness constants and other required parameters are taken from Miles et al. [2] work as listed in Table 1. We apply a pressure of 1 Pa having 45° incidence angle, where the left side is at leading end. To create an analogy with a real world scenario, the angle of incidence θ is used in the following equation ϕ ¼ 2π

d sin θ λ

(2)

where d is the model center separation, ϕ is the initial phase difference and λ is the sound source wavelength [18]. The wavelength is accounted for individual frequency step of testing. The first analysis is done by applying the critical damping ratio (ζ¼1) as illustrated in Fig. 2(a). The solid line represents the leading side and the dotted line is for the lagging side. The leading side is constantly moving at higher magnitude than the lagging side. The directionality can be easily estimated by this magnitude difference. The magnitude difference is much prominent in the range of 4–30 KHz because it exists between the two modes of vibration. The results are proving the effect of critical damping in enhancing the directional sensitivity cues. The second analysis is done by considering the overdamped system (ζ ¼10) as depicted in Fig. 2(b). The amplitude of vibration is very small. The left side is moving with bit higher magnitude than the right side but the magnitude difference is very small, and sound sensitivity cues extraction are very challenging. The overdamped case also nullifies the highlights of rocking and bending modes as depicted in previous critical damping case. Fig. 2(c) is the output of underdamped (ζ¼0.1) case. The two modes of vibration are very clear. The rocking mode is around 6 KHz and the bending mode is around 31 KHz. The magnitude of vibration is very high, however, the amplitude difference is negligible, making the model less sensitive to direction localization. Fig. 2(d) is showing the output of undamped (ζ ¼0) case. The results are very similar to the underdamped case, and both modes are quite prominent. Although the magnitude of vibration is very high, the case fails to give cues for the directional sensitivity. To check the directivity response of the model on both on-axis and off-axis angles of incidence, we put angle of incidence of 0°, 30°, 60° and 90° in Eq. (2), while setting the critical damping ratio (ζ ¼1). In Fig. 3(a), the zero incidence angle has roughly insignificant amplitude difference on both sides. The amplitude difference increases with the increase of angle of incidence and reaches to its maximum at 90° incidence angle as shown in Fig. 3. Both rocking and bending modes are quite detectable in the last three cases, but the rocking mode is not shown in the first case. These outcomes are confirming the bidirectional response of the mechanical model, which is a desired response for the directional microphones. All these results support the concept that a careful modeling of squeeze air film damping can help us to better predict the sound source localization cues. The key to get this sensitivity accuracy is to bring the damping ratios of both modes in the range of critical damping. To validate these encouraging results, we perform an analytical modeling of squeeze air film damping of both modes of vibration. The derived damping formulas are analyzed with different values of the hole radius and the initial gap thickness to find the damping ratios close to the critical damping. Table 1 List of parameters and dimensions used in mathematical model and in diaphragm design. Symbol m d K1 ¼ K2 Kt L W Tp Lp h0 NH Pa μ ρ E

Description Mass of the model Model center separation Stiffness of both sides of the model Stiffness of torsional beams of the model Length of the diaphragm Width of the diaphragm Thickness of the diaphragm Pitch of holes Initial gap thickness Total number of holes Ambient pressure Viscosity of air Density of silicon Young's modulus of silicon

Values

Unit  10

2.88  10 1200 0.576 5.18 1400 1100 4.0 160 11 63 101,325 1.83  10  5 2328 130  109

kg μm N m1 N m1 μm μm μm μm μm N m2 N s m2 kg m  3 N m2

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Fig. 2. Frequency response of the model on different damping ratios: (a) damping ratio¼1, (b) damping ratio¼ 10, (c) damping ratio¼0.1 and (d) damping ratio¼0.

3. Rocking mode The oscillations of both sides of the diaphragm at 180° out of phase are called rocking mode. In this mode, the structure has see-saw type motion. The rocking mode of the microphone is shown in Fig. 4(a). The length and the width of the diaphragm are represented by L and W respectively. The nominal gap thickness between the diaphragm and the substrate is h0 . The rotation of the diaphragm is θ. The radius of the hole is r, and the pitch of holes is Lp , as shown in the cross-section view of the rocking mode in Fig. 4(b). The modified Reynolds equation of squeeze air film damping presented by Pandey et al. [10] is used for analysis. The equation for circular holes is ( ! !) 3 3 ∂ PQ ch h ∂P ∂ PQ ch h ∂P Q β2 r 2 ∂ðPhÞ (3) P ðP  P a Þ ¼ þ  th ∂x ∂y ∂t 12μ ∂x 12μ ∂y 8μT eff ηðβÞ where Pðx; y; tÞ is the air pressure in gap. β ¼ r=r 1 is the perforation ratio, r is the radius of the hole and r 1 ¼ 0:525lp is the radius of the cell. Teff ¼Tp þ(3πr/8) is the effective hole length, which includes the hole length Tp and an equivalent length to account for the end effect of hole. h(t) is the squeeze air film thickness and μ is the air viscosity. 3 ηðβÞ ¼ ð1 þ ð3r 4 KðβÞQ th =16T eff h Q ch ÞÞ and KðβÞ ¼ 4β2  β4  4 ln β  3. Qch and Qth are the flow rate factors, which account for rarefaction effect in the flow through the parallel plates and through the holes respectively. Q th ¼ 1 þ 4Knth and expression for Qch is given by [10] pffiffiffi 0:01807 π 1:35355 þ 6 1:17468 (4) Q ch ¼ 1 þ3 D0 D0 pffiffiffi where D0 ¼ π =2Knth ; Knch ¼ λ=h; Knth ¼ λ=r and λ ¼ 0:0068=P a at ambient temperature and pressure Pa. In rocking mode, by

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Fig. 3. Frequency response of the model on different angle of incidence considering critical damping ratio: (a) 0°, (b) 30°, (c) 60°, and (d) 90°.

considering θ0 ejwt as the angular displacement of the diaphragm, the squeeze air film thickness variations can be represented by hðx; t Þ ¼ h0 þ x θ0 ejωt ;

L L  rxr 2 2

(5)

Eq. (3) can be linearized by considering small amplitude of vibration ðh0 ⪢x  θ0 Þ, where h ¼ h0 þ Δh and small pressure variation ðP a ⪢ΔPÞ, where P ¼ P a þ ΔP. For simplicity, we introduce some non-dimensional variables P¼

ΔP Pa

H ¼ d0 ejωt

(6)

where d0 ¼ θ0 =h0 . The linearized modified Reynolds equation as per substitution of non-dimensional variables is ∂2 P ∂2 P P ∂P ∂H þ  ¼ α2 þ α2 x ∂t ∂t ∂x2 ∂y2 l2

(7)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 2 2 where l ¼ 2h0 T eff ηðβÞQ ch =3β2 b Q th and α2 ¼ 12μ=h0 P a Q ch 3.1. Green's function solution

Eq. (7) is a linear, nonhomogeneous partial differential equation. It is hard to directly solve the equation, so we transform Eq. (7) to the standard 2D diffusion equation, and after that we solve it by using Green's function. For the transformation, following functions are being used 2

P ðx; y; t Þ ¼ U ðξ; γ; t Þe  k t ;

ξ ¼ x; γ ¼ y; k ¼

1 lα

(8)

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Fig. 4. (a) Rocking mode. (b) Cross-section view of the rocking mode.

by putting the above transformation functions and rearranging, Eq. (7) becomes ∂2 U ∂2 U ∂U ¼ f ðx; y; t Þ þ  α2 ∂t ∂x2 ∂y2

(9)

2

where f ðx; y; t Þ ¼ xα2 ek t ∂H=∂t. The transformation does not affect x and y and they are same. Eq. (9) is a linear 2D diffusion equation with a source term f ðx; y; tÞ and can be solved easily by using Green's function. For boundary conditions, pressure on the boundaries is taken zero and for initial conditions ðPðt ¼ 0Þ ¼ 0Þ. By applying Green's function, we solve Eq. (9) for Uðx; y; tÞ U ðx; y; t Þ ¼

mπx nπy X 8Lð 1Þðm þ n  2Þ=2  jωd0 ejwt ek2 t cos sin 2 2 L W nmπ m ¼ even k 2 n ¼ odd k þ mn þjω α2 2

The equation can be transformed into Pðx; y; tÞ; by putting U ðx; y; t Þ ¼ P ðξ; γ; t Þek P ðx; y; t Þ ¼

(10)

t

mπx nπy X 8Lð  1Þðm þ n  2Þ=2 jωd0 ejwt sin cos 2 2 L W nmπ m ¼ even k 2 n ¼ odd k þ mn þ jω α2

(11)

where 2

2

2

kmn ¼ km þ kn ¼

m2 π 2 2

L

þ

n2 π 2 W2

(12)

The restoring torque TðtÞ on the oscillating diaphragm is calculated by performing integration of the pressure distribution over the domain f  L=2 r x r L=2;  W=2 r yr W=2g. Let A be equal to the constant terms that are not participating in integration A¼

8Lð  1Þðm þ n  2Þ=2  jωd0 ejwt 2 nmπ 2 k 2 k þ mn þ jω α2

(13)

by performing integration T ðt Þ ¼

X

Z A

m ¼ even n ¼ odd

W 2

W 2

Z

L 2

 2L

P a x sin

mπx nπy X  2P a L2 Wð  1Þðm þ n  2Þ=2 cos dx dy ¼ A L W mnπ 2 m ¼ even n ¼ odd

(14)

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Fig. 5. (a) Bending mode. (b) Cross-section view of the bending mode.

The total torque including constant terms is as T ðt Þ ¼

X 16P a L3 Wð  1Þðm þ n  2Þ  jωd0 ejwt 2 n 2 m2 π 4 m ¼ even k 2 n ¼ odd k þ mn þ jω α2

The absolute value of the damping torque Td is calculated by taking the imaginary part of the total torque TðtÞ ! 2 kmn 2 ωd k þ 0 X 16P a L3 Wð  1Þðm þ n  2Þ α2 Td ¼ !2 2 2 4 2 n m π m ¼ even k 2 n ¼ odd k þ mn þ ω2 α2

(15)

(16)

The absolute value of the spring torque Ts is calculated by taking the real part of the total torque T(t) Ts ¼

X 16P a L3 Wð  1Þðm þ n  2Þ n 2 m2 π 4 m ¼ even n ¼ odd

ω2 d0 !2 2 kmn 2 k þ 2 þω2 α

The damping constant Cr for the rocking mode caused by the pressure of the squeeze air film is given by ! 2 kmn 2 k þ X 16P a L3 Wð 1Þðm þ n  2Þ α2 T Cr ¼ d ¼ !2 2 2 4 θ0 ω m ¼ even 2 n m π h0 k 2 n ¼ odd þω2 k þ mn α2

(17)

(18)

The spring constant K r for the rocking mode caused by the pressure of the squeeze air film is given by Kr ¼

X 16P a L3 Wð 1Þðm þ n  2Þ Ts ¼ θ0 m ¼ even n2 m2 π 4 h0 n ¼ odd

ω2 2 k 2 k þ mn α2

(19)

!2 þ ω2

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The damping ratio ζ r is calculated by the following expression ζr ¼

Cr 2ωI r

(20)

where Ir is the mass moment of inertia and it is calculated as 

   r2 1 2 þdi Mi I r ¼ ρL3 WT p  πr 2 ρT p 12 4

(21)

where Mi represents the total number of holes at the distance di from the center.

4. Bending mode The in-phase oscillations of both sides of the diaphragm are called the bending mode. In this mode, both sides of the diaphragm move up and down at the same time. The bending mode of the microphone is shown in Fig. 5(a). The crosssection view is also illustrated in Fig. 5(b). As the oscillations of both left and right sides of the microphone is identical, so the most ideal approach to solve this motion is to divide the structure into two parts from the center, and solve the left side first and then multiply the results by 2 to account for the right side. In bending mode, by considering the half length of the diaphragm as Lx , the squeeze air film thickness variations can be expressed as hðx; t Þ ¼ h0 þ x θ0 ejωt ;

0 r x r Lx

(22)

The pressure distribution under the oscillating diaphragm in the bending mode can be obtained by following the same steps as mentioned in the rocking mode. The value of the pressure can be expressed as   2 ðn  1Þ=2 ðm  1Þ=2

8L ð  1Þ ð  1Þ  x nπy X jωd0 ejwt mπx mπ (23) cos cos P ðx; y; t Þ ¼ 2 2 2Lx W mnπ kmn m ¼ odd 2 n ¼ odd k þ 2 þ jω α where 2

2

2

kmn ¼ km þ kn ¼

m2 π 2 4L2x

þ

n2 π 2

(24)

W2

The restoring torque TðtÞ on the oscillating diaphragm is calculated by performing integration of the pressure distribution over the domain f0 r x rLx ;  W=2 r yr W=2g. Let A be equal to the constant terms that are not participating in integration as   2 8Lx ð  1Þðn  1Þ=2 ð  1Þðm  1Þ=2  jωd0 ejwt mπ A¼ (25) 2 mnπ 2 k 2 k þ mn þ jω α2 by performing integration

T ðt Þ ¼

X m ¼ odd n ¼ odd

Z A

W 2

W 2

Z

Lx

0

  n1 m1 2 2 2 2 

4L Wð 1Þ ð 1Þ nπy x X mπx mπ dx dy ¼ P a x cos A cos 2Lx W mnπ 2 m ¼ odd

(26)

n ¼ odd

The total torque becomes

T ðt Þ ¼

  2 2 3 ðm  1Þ=2 64P L W ð 1Þ  a x X mπ m ¼ odd n ¼ odd

n 2 m2 π 4

jωd0 ejwt 2

k þ

2

kmn þ jω α2

:

The absolute value of the damping torque Td is calculated by taking the imaginary part of the total torque T(t) ! 2   2 2 ωd k2 þ kmn 3 ðm  1Þ=2 0  X 64P a Lx W ð  1Þ α2 mπ Td ¼ !2 2 m2 π 4 2 n m ¼ odd k 2 n ¼ odd þ ω2 k þ mn α2

(27)

(28)

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The absolute value of the spring torque Ts is calculated by taking the real part of the total torque T(t)   2 2 3 ðm  1Þ=2  X 64P a Lx W ð  1Þ ω2 d 0 mπ Ts ¼ !2 2 2 4 2 n m π m ¼ odd k 2 n ¼ odd k þ mn þ ω2 α2

(29)

The damping constant Cb1 for the bending mode of half of the diaphragm caused by the pressure of the squeeze air film is given by ! 2  2 kmn 2 2 3 ðm  1Þ=2 k þ 2  X 64P a Lx W ð 1Þ α T mπ (30) C b1 ¼ d ¼ !2 θ0 ω m ¼ odd 2 n2 m2 π 4 h0 kmn 2 n ¼ odd 2 k þ 2 þω α The spring constant K b1 for the bending mode of half of the diaphragm caused by the pressure of the squeeze air film is given by   2 2 64P a L3x W ð 1Þðm  1Þ=2  X Ts ω2 mπ K b1 ¼ ¼ (31) !2 θ0 m ¼ odd 2 n2 m2 π 4 h0 kmn 2 n ¼ odd 2 k þ 2 þω α Thus, the damping constant of the whole diaphragm is given by C b ¼ 2:C b1 . The damping ratio ζ b is calculated by the following expression ζb ¼

Cb 2ωI b

where Ib is the mass moment of inertia and it is calculated as 

   r2 1 2 þdi I b ¼ ρL3 WT p  πr 2 ρT p Mi 12 4

(32)

(33)

where Mi represents the total number of holes at the distance di from the center.

5. Validation and discussions The intention to drive an analytical model of the perforated directional microphone is to show the usefulness of derived damping formulas to optimize the microphone parameters, namely, hole radius and initial gap thickness, to achieve the critical damping. We bring variations in the individual parameters and also explain their impacts on the squeeze air film damping behavior. The validation of the analytical modeling results is shown by comparing their accuracy with Li et al. [13] work and finally with FEM results. The parameters and dimensions utilized as a part of the investigation are listed in Table 1. 5.1. Parameters optimization for estimating critical damping ratios The analysis is done by changing the individual parameters in the damping formulas to get the damping ratio in the vicinity of the critical damping. The pitch of holes is varied from 60 μm to 160 μm by changing the hole radii span from 10 μm to 40 μm. The radius of hole is not considered above 40 μm as to avoid issues for acoustics actuation, since bigger holes can influence the amount of driving pressure. The gap thickness is varied from 5 μm to 11 μm. The outcomes are shown in Fig. 6. 5.1.1. Effect of hole radius The damping ratios are inversely related to the hole radius. The response gets closer to overdamped as the hole radius decreases, and becomes underdamped as the hole radius increases, for the same gap thickness and pitch of holes. The reason behind this is that as the hole size is getting smaller, the air has less passage to escape from the diaphragm, thus it puts more pressure on the diaphragm and makes it overdamped. The explanation for the system to become underdamped is likewise evident because now the air has more area to leave the system and effect decreases. This can be observed in Fig. 6. 5.1.2. Effect of gap thickness The gap thickness also controls the amount of air to leave or enter the system. It also demonstrates an inverse relation with the damping ratios. Increase and decrease in the gap thickness moves the system towards the overdamped and underdamped cases respectively by keeping the remaining conditions same. The explanation of this behavior is very obvious

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Fig. 6. Effect of parameters variation on damping ratios: (a) rocking mode at g ¼ 5 μm, (b) bending mode at g ¼5 μm, (c) rocking mode at g ¼ 8 μm, (d) bending mode at g¼ 8 μm, (e) rocking mode at g ¼11 μm, and (f) bending mode at g ¼ 11 μm.

because as the gap thickness increases, the air gets more space to leave the system, so the pressure decreases and the damping ratio decreases. On the other hand, decreases in the gap thickness entrap the air, so it puts more pressure and the damping ratio increases. The effects can be observed in Fig. 6.

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The damping ratio of the rocking mode is approximately at the critical damping value at a radius of 20 μm, pitch of holes of 160 μm and initial gap thickness of 11 μm. The corresponding value of the bending mode is 0.4842. Both values can be depicted in Fig. 6(e) and (f) respectively. The approximate critical damping ratio of the bending mode can be observed at a radius of 20 μm, pitch of holes of 160 μm and initial gap thickness of 8 μm. The corresponding value of the rocking mode for these parameter values is 2.1690 and both values can be seen in Fig. 6(d) and (c) respectively. From Section 2, we observe that taking the damping ratio higher than the critical damping affects the system badly because it diminishes the highlights of modes. Furthermore, the vibration magnitude is low and extremely hard to detect. On the other hand, the damping ratio below the critical damping highlights the two modes, and also the vibration is very high. To design in real life cases, it is ideal to pick the damping ratio that is exceptionally close to the critical damping, however, it can be acceptable if bit lower than it because it has high extent of vibration. Therefore, the best case from the above discussion is that to choose parameter values that make the rocking mode in the critical range as highlighted in Fig. 6. 5.2. Comparison with Li et al. work In 2015, Li et al. [13] presented an analytical model of the squeeze air film damping of the rocking mode of a torsional micro-resonator. They used a rectangular structure with square shaped perforations. They solved the pressure underneath the vibrating structure using double sine series. The damping constant equation of their analysis is "

2 #  1 P a σ X 16L3 W L 2 L 2 C Iθ ¼  ðmπÞ þ nπ: þ (34) g 0ω m ¼ evenm2 n2 π 4 W l n ¼ odd

where σ is the squeeze number. For the comparison of results, we have computed the damping constants on different values of the hole radius. The output shows a very good agreement between these two results as listed in Table 2. There is no such discrepancy in both results because both techniques end with approximate same equation. 5.3. Finite element method analysis FEM is the most accurate way to predict the squeeze air film damping analysis of vibrating miniature structures. We design a rectangular perforated diaphragm in ANSYS software by using the parameter values as listed in Table 1. The hole radius value is 20 μm. First, we perform a modal analysis on the structure to observe the first two fundamental modes of vibration. The analysis shows the same expected modes. The rocking mode appears at first natural frequency, where both sides are at 180° out of phase as shown in Fig. 7(a). The bending mode appears at second natural frequency, where both sides are moving in-phase as shown in Fig. 7(b). The structure has maximum deflection at the boundaries and minimum at the center. FLUID 136 and FLUID 138 elements are used to predict the damping ratios. The FLUID 136 element is used to model the fluid domain between the vibrating diaphragm and the fixed substrate, while the FLUID 138 element is used for the pressure drop through the circular holes as illustrated in Fig. 8. FLUID 136 and FLUID 138 elements require mean free path and ambient pressure for proper functioning. The one dedicated parameter for FLUID 136 element is the initial gap thickness, which caters for the amount of back volume. Similarly, the one dedicated parameter for FLUID 138 is the hole radius, which caters for the amount of open area. The pressure on the boundaries is kept zero. For simplicity and comparison purposes, the gratings are not included during the FEM analysis because they are negligible in size compared to the total size of the diaphragm. The comparative analysis of rocking and bending modes damping ratios of analytical and FEM results is listed in Table 3. The mode frequencies used in the analysis are also listed. The analytical results are showing good agreement with the FEM results. The minor differences in the damping ratio results are because, in analytical model, we consider the gap variations of the structure in the z-direction only, while ANSYS caters for the gap variations in the z-direction and also to the minor variations in the y-direction. These minor variations affect the damping ratio results. Secondly, in analytical model there is no Table 2 Comparative analysis of the damping constants of the rocking mode. Hole radius (μm) Damping constants by Green's function (N m s rad  1) Damping constants by double sine series (N m s rad  1) Percentage error 65 60 55 50 45 40 35 30

9.897  10  12 2.360  10  11 4.650  10  11 8.082  10  11 1.287  10  10 1.927726  10  10 2.756153  10  10 3.810495  10  10

9.897  10  12 2.360  10  11 4.650  10  11 8.082  10  11 1.287  10  10 1.927727  10  10 2.756154  10  10 3.810496  10  10

0.0 0.0 0.0 0.0 0.0 0.00005 0.00004 0.00003

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433

parameter that deals with the torsional beams specifications. On the other hand, in ANSYS, we design two torsional beams having 10 μm length, 22 μm width and 4 μm thickness. These two beams are fixed on the opposite sides. Although the dimensions of the torsional beams are very small, they are capable of bringing some minor changes in the results. A harmonic analysis is performed in ANSYS from 0 to 10 KHz frequency range. A pressure of 1 Pa having 45° phase difference, where the left side is considered at a leading end, is applied on the designed microphone. To capture the dynamics of the

Fig. 7. FEM derived modes of vibration: (a) rocking mode and (b) bending mode.

Fig. 8. Fluid elements used in ANSYS analysis.

Table 3 Comparative analysis of damping ratios of analytical model and ANSYS results. Hole radius (μm) Gap thickness (μm)

Modal frequency (Hz)

Damping ratio, Rocking mode (Analytical model)

Damping ratio, Rocking mode (ANSYS)

20

3730

0.9916

1.0913

Damping ratio, Bending mode (Analytical model)

Damping ratio, Bending mode (ANSYS)

0.4842

0.5824

20

11

11

7657

434

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Fig. 9. Frequency response of both sides of the diaphragm: (a) g ¼11 μm, (b) g ¼ 5 μm, (c) g¼ 40 μm, and (d) zero damping.

holes in loading, the fluid elements are used as part of investigation. The analysis is performed for the gap thickness of 11 μm, 5 μm and 40 μm, to investigate their effects on the amplitude of vibration and to validate the mathematical modeling results as shown in Fig. 9. The gap thickness of 11 μm is the desired case, and the results are very encouraging because they show a significant amplitude difference on both sides. The amplitude difference is maximum in the middle of rocking and bending modes frequencies. The magnitude difference diminishes exceptionally fast after the bending mode frequency as shown in Fig. 9(a). The gap thickness of 5 μm moves the diaphragm response towards the overdamped case. The two modes are diminished and the amplitude of vibration is very minute. The two sides are moving approximately at the same magnitude, thus nullifying the directional sensitivity cues as shown in Fig. 9(b). The gap thickness of 40 μm makes the diaphragm underdamped with a very clear two modes. The rocking mode is around 3.7 KHz frequency and the bending mode is around 7.6 KHz frequency. In spite of the fact that the amount of vibration is entirely huge, the case fails to deliver a significant amplitude difference on both sides as shown in Fig. 9(c). One further investigation is performed by applying the zero damping on the diaphragm, while keeping the remaining conditions same. The results are pointing the two vibration modes along with very high magnitude of vibration. The relative amplitude difference is fairly negligible in the entire range of the frequency as shown in Fig. 9(d). This design configuration is suitable for sound source localization up to 8.0 KHz frequency. All these results are supporting the idea of critical damping parameter effects in enhancing the directional sensitivity cues by creating a significant amplitude difference on both sides. Since, the increased damping will raise the noise floor of the microphone signal [20], noise analysis is required to estimate its usefulness as a practical sensor. However, this work opens up the possibility of tuning of the damping, which can help us to design a directional microphone mimicking the fly's ear system.

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6. Conclusion We present an analytical model of squeeze air film damping of the directional microphone. First, we present our unique perforated biomimetic rectangular shaped microphone design, and by mathematical modeling, we prove that if we can bring the damping ratio of the microphone in the critical damping range then the microphone can produce a very good directional sensitivity cues. Analytically, we derive the squeeze air film damping formulas for first two fundamental modes of vibration of the microphone using Green's function. The microphone parameters like hole radius and gap thickness are varied in derived damping formulas to locate the damping ratios of both modes in the vicinity of critical damping ratio. The directional microphone is designed in ANSYS on the basis of derived parameters. The analytical modeling results are validated by comparing with previous renowned work done in the rocking mode and also with the FEM results. The analytical results show an excellent agreement in both validations. The FEM simulation results of the directional microphone also demonstrate the positive effect of critical damping ratio in enhancing the directional sensitivity cues for incoming sound. Acknowledgment This work is supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 2013R1A1A2007684).

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