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Research Papers

Thermoelastic Damping in Asymmetric Three-Layered Microbeam Resonators

[+] Author and Article Information
Wanli Zuo, Jianrun Zhang

School of Mechanical Engineering,
Southeast University,
Nanjing 211189, China

Pu Li

School of Mechanical Engineering,
Southeast University,
Nanjing 211189, China
e-mail: seulp@seu.edu.cn

Yuming Fang

College of Electronic Science and Engineering,
Nanjing University of Posts and Telecommunications,
Nanjing 210023, China

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 8, 2015; final manuscript received March 3, 2016; published online March 28, 2016. Assoc. Editor: George Kardomateas.

J. Appl. Mech 83(6), 061002 (Mar 28, 2016) (15 pages) Paper No: JAM-15-1419; doi: 10.1115/1.4032919 History: Received August 08, 2015; Revised March 03, 2016

Thermoelastic damping (TED) has been recognized as a significant mechanism of energy loss in vacuum-operated microresonators. Three-layered microbeams are common elements in many microresonators. However, only the model for TED in the three-layered microbeams with symmetric structure has been developed in the past. The first and the third layers in these beams have the same thickness and material properties. Thus, the temperature field is symmetric in these beams. In this paper, an analytical expression for TED in the asymmetric three-layered microbeams is developed in the form of an infinite series. The temperature fields in the asymmetric three-layered microbeams are asymmetric. The total damping is obtained by computing the energy dissipated in each layer. It is seen that the values for TED computed by the present model agree well with those computed by the finite-element model. The limitations of the present model are assessed. A simple model is also presented by retaining only the first term. The accuracy of the simple model is also discussed. The present model can be used to optimize the design of three-layered microbeams.

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Figures

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Fig. 1

A schematic cross section of a symmetric, three-layered beam and the coordinate system

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Fig. 2

A schematic cross section of a bilayered beam and the coordinate system

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Fig. 3

A schematic cross section of a three-layered Euler–Bernoulli beam and the coordinate system

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Fig. 4

Convergence analysis of the present model in Ag/Si/Ag with h1 = 0.5 μm, h2 = 9 μm, and h3 = 1 μm

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Fig. 5

Comparison of TED obtained by the present model and the ansys model for four fixed–fixed beams (Ag/Si/Ag). The four beams have different thicknesses of the third layer.

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Fig. 6

Comparison of TED obtained by the present model and ansys model for four fixed–fixed beams (Au/SiC/Au). The four beams have different thicknesses of the second layer.

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Fig. 7

Comparison of TED obtained by the present model and the ansys model in metallized silicon beams

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Fig. 8

Comparison of TED obtained by the present model and ansys model for four cantilever beams (Cu/SiC/Cu). The four beams have different thicknesses of the third layer.

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Fig. 9

Comparison of TED obtained by the present model and the ansys model for four fixed–fixed microbeams (Ag/Si/Ag). The four microbeams have different lengths.

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Fig. 10

The imaginary parts of the temperature variation in the fixed–fixed microbeam

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Fig. 11

The imaginary parts of the temperature variation in the cantilever microbeam. The cantilever is clamped at the edge x = 0.

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Fig. 12

The effect of metallization on TED in Si microbeams

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Fig. 13

Convergence analysis of the present model in the case of SiO2/Si/Zn microbeam

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Fig. 14

The value of qn as a function of n in the case of SiO2/Si/Zn microbeam

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