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Research Papers

Passive Regulation of Thermally Induced Axial Force and Displacement in Microbridge Structures

[+] Author and Article Information
Pezhman Hassanpour

Mem. ASME
Assistant Professor
Department of Mechanical Engineering,
Loyola Marymount University,
Los Angeles, CA 90045
e-mail: phassanpour@lmu.edu

Patricia M. Nieva

Professor
Department of Mechanical and
Mechatronics Engineering,
University of Waterloo,
Waterloo, ON N2 L 3G1, Canada

Amir Khajepour

Professor
Mem. ASME
Department of Mechanical and
Mechatronics Engineering,
University of Waterloo,
Waterloo, ON N2 L 3G1, Canada

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 8, 2017; final manuscript received September 14, 2017; published online October 12, 2017. Assoc. Editor: M Taher A Saif.

J. Appl. Mech 84(12), 121003 (Oct 12, 2017) Paper No: JAM-17-1430; doi: 10.1115/1.4037933 History: Received August 08, 2017; Revised September 14, 2017

The analytical model of a mechanism for regulating the thermally induced axial force and displacement in a fixed–fixed microbeam is presented in this article. The mechanism which consists of a set of parallel chevron beams replaces one of the fixed ends of the microbeam. The thermomechanical behavior of the system is modeled using Castigliano’s theorem. The effective coefficient of thermal expansion is used in the analytical model. The analytical model takes into account both the axial and bending deformations of the chevron beams. The model provides a closed-form equation to determine the thermally induced axial force and displacement in the microbeam. In addition, the model is used to derive the equations for the sensitivities of the microbeam’s axial force and displacement to the variations of the design parameters involved. Moreover, the model produces the stiffness of the chevron beams. The effect of the stiffness of the chevron beams on the dynamic behavior of the microbeam is discussed. The analytical model is verified by finite element modeling using a commercially available software package. Using the analytical model, two special cases are highlighted: a system with thermally insensitive axial force and a system with thermally insensitive axial displacement. The main application of the model presented in this article is in the design of sensors and resonators that require robustness against changes of temperature in the environment. The analytical model and the sensitivity equations can be easily integrated into optimization algorithms.

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Figures

Grahic Jump Location
Fig. 1

Top: scanning electron microscope picture of a microbridge with chevron structure attached to its right end. This microbridge is designed for sensing applications [30]. The microbridge is in the gap between two electrodes for electrostatic excitation and sensing. Bottom: the schematic of the microbridge and thermal stress regulator. The microbridge is attached to the hub of the chevron structure, which is free to move in the axial direction. The electrostatic electrodes are not shown.

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Fig. 2

Two parallel bars/beams with equal lengths fixed at the left end and welded to a rigid plate at right

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Fig. 3

Left: boundary conditions of a single fishbone beam. The fixed end (upper end for the beam shown) represents the anchored ends of chevron beams. The guided end (lower end here) is the point that the beam is connected to the hub. Right: reaction forces at the guided end of the fishbone beam. The rotation of the beam at this boundary is assumed to be zero; therefore, a bending moment M, in addition to reaction forces P and Q, is applied to the beam.

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Fig. 4

Reaction force applied to the sensing beam from the set of chevron beams

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Fig. 5

Simplified model of the modified boundary conditions of the sensing beam

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Fig. 6

Axial stress (left) and hub-end axial displacement (right) of the sensing beam of the sample device 1 versus temperature using the analytical model (solid line) and FEM model (data marks)

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Fig. 7

Axial stress (left) and hub-end axial displacement (right) of the sensing beam of the sample device 2 versus temperature using the analytical model (solid line) and FEM model (data marks)

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Fig. 8

Axial stress (left) and hub-end axial displacement (right) of the sensing beam of the sample device 3 versus temperature using the analytical model (solid line) and FEM model (data marks)

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Fig. 9

Axial stress (left) and hub-end axial displacement (right) of the sensing beam of the sample device 4 versus temperature using the analytical model (solid line) and FEM model (data marks)

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Fig. 10

Axial stress (left) and hub-end axial displacement (right) of the sensing beam of the sample device 5 versus temperature using the analytical model (solid line) and FEM model (data marks)

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Fig. 11

Graphic representation of Eq. (16). The axial force of the sensing beam becomes thermally insensitive by selecting a combination of parameters from this design map. Each line in this figure corresponds to the angle shown on it in degrees.

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Fig. 12

Sensitivity ∂2p/∂ν∂τ with respect to ηc for several β’s assuming ∂p/∂τ=0. In this figure, N = 4, μ = 1, and ηs=1/480,000. The equivalent range of ν is different for each curve.

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Fig. 13

Graphic representation of Eq. (25). The axial displacement of the sensing beam becomes thermally insensitive by selecting a combination of parameters from this design map.

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