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Research Papers

# On the Equivalence of Acoustic Impedance and Squeeze Film Impedance in Micromechanical Resonators

[+] Author and Article Information
Charanjeet Kaur Malhi

Centre for Ultrasonic Engineering,
Department of Electronic
and Electrical Engineering,
University of Strathclyde,
Glasgow G1 1XW, UK
e-mail: charanjeet.malhi@gmail.com

Rudra Pratap

Professor
Department of Mechanical Engineering
and Centre for Nano Science and Engineering,
Indian Institute of Science,
Bangalore 560012, India
e-mail: pratap.mems@gmail.com

1Corresponding author.

Contributed by the Noise Control and Acoustics Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received April 4, 2015; final manuscript received August 14, 2015; published online October 15, 2015. Assoc. Editor: Sheryl M. Grace.

J. Vib. Acoust 138(1), 011005 (Oct 15, 2015) (10 pages) Paper No: VIB-15-1111; doi: 10.1115/1.4031438 History: Received April 04, 2015; Revised August 14, 2015

## Abstract

In this work, we address the issue of modeling squeeze film damping in nontrivial geometries that are not amenable to analytical solutions. The design and analysis of microelectromechanical systems (MEMS) resonators, especially those that use platelike two-dimensional structures, require structural dynamic response over the entire range of frequencies of interest. This response calculation typically involves the analysis of squeeze film effects and acoustic radiation losses. The acoustic analysis of vibrating plates is a very well understood problem that is routinely carried out using the equivalent electrical circuits that employ lumped parameters (LP) for acoustic impedance. Here, we present a method to use the same circuit with the same elements to account for the squeeze film effects as well by establishing an equivalence between the parameters of the two domains through a rescaled equivalent relationship between the acoustic impedance and the squeeze film impedance. Our analysis is based on a simple observation that the squeeze film impedance rescaled by a factor of $jω$, where ω is the frequency of oscillation, qualitatively mimics the acoustic impedance over a large frequency range. We present a method to curvefit the numerically simulated stiffness and damping coefficients which are obtained using finite element analysis (FEA) analysis. A significant advantage of the proposed method is that it is applicable to any trivial/nontrivial geometry. It requires very limited finite element method (FEM) runs within the frequency range of interest, hence reducing the computational cost, yet modeling the behavior in the entire range accurately. We demonstrate the method using one trivial and one nontrivial geometry.

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## Figures

Fig. 1

MEMS resonator with various zones of dissipation—squeeze film action in between the resonator and fixed substrate and energy dissipation in the infinite fluid on top of the resonator due to the acoustic waves

Fig. 2

Comparative between acoustics and sqds: (a) acoustic impedance, (b) squeeze film impedance (original), and (c) modified squeeze film impedance

Fig. 3

Equivalent circuit for the acoustic domain with its high- and low-frequency behaviors

Fig. 4

Pressure distribution obtained from the FEM model for a typical frequency of oscillation of the plate

Fig. 5

FLUID138 element modeling

Fig. 6

Plate with slits and holes and its SEM image

Fig. 7

Rectangular plate with a rectangular slot modeled in two ways—using a single FLUID138 element and using four FLUID138 elements

Fig. 8

Various steps in determining the expressions for the equivalent elements of the sqd using the circuit valid in acoustic domain

Fig. 9

Low-frequency real impedance for determination of mass in the acoustics domain which is equivalent to resistance loading in the sqd. The lowest value of σ on the X axis is 0.003.

Fig. 10

High-frequency imaginary impedance for determination of resistance in the acoustics domain which is equivalent of stiffness in the sqd

Fig. 11

Low-frequency imaginary impedance for determination of total resistance in the acoustics domain in the low-frequency regime which is equivalent to stiffness in the sqd. The lowest value of σ2 on the X axis is 9 ×  10−6.

Fig. 12

Acoustic impedance (which is equal to the squeeze film impedance multiplied by jω) obtained using curvefit and analytical formulation for the plate without slits. The relevant dimensions for the calculation of the damping and the stiffness impedances analytically are: h0 = 2 μm, η = 1 for the isothermal process, μ = 1.85 × 10−5 Pa·s, a = 295 μm, hdash = 0.1 (displacement assumed is one-tenth of the original gap), and V = ωhdash.

Fig. 13

Acoustic impedance (which is equal to the squeeze film impedance multiplied by jω) obtained using curvefit and analytical formulation for the plate with slits

Fig. 14

Normalized forces in the original sqd obtained using curvefit and analytical formulation for the plate with slits

Fig. 15

Acoustic impedance divided by ω (which is equal to the squeeze film impedance) obtained using curvefit for the plate with slits

Fig. 16

Acoustic impedance divided by ω (which is equal to the squeeze film impedance) obtained using curvefit for the solid plate without slits. The range of applicability is extended beyond the range of values used for curvefit procedure.

Fig. 17

Equivalent circuit for a resonator. The effect due to the radiation impedance, electrostatic stiffness is ignored for analysis purpose.

Fig. 18

Simulated frequency response of the simulator using the squeeze film parameters derived from our curvefit and using the squeeze film damping and stiffness forces from Darling et al. [5]. The relevant dimensions for the calculation of the acoustic compliance and mass are: a = 295 μm, ρ = 2320 kg/m3, h = 2 μm, E = 160 GPa, and ν = 0.22. The relevant dimensions for the calculation of the damping and the stiffness impedances analytically are: h0 = 2 μm, η = 1 for the isothermal process, μ = 1.85 × 10−5 Pa·s, hdash = 0.1(displacement assumed is one-tenth of the original gap), and V =  ωhdash.

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