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Research Papers

Voltage-Induced Snap-Through of an Asymmetrically Laminated, Piezoelectric, Thin-Film Diaphragm Micro-Actuator—Part 2: Numerical and Analytical Results

[+] Author and Article Information
W. C. Tai

Department of Mechanical Engineering,
Virginia Polytechnic Institute and
State University,
Blacksburg, VA 24061
e-mail: wchtai@vt.edu

I. Y. Shen

Professor
Department of Mechanical Engineering,
University of Washington,
Seattle, WA 98195-2600

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 24, 2017; final manuscript received February 22, 2018; published online April 17, 2018. Assoc. Editor: Matthew Brake.

J. Vib. Acoust 140(5), 051006 (Apr 17, 2018) (10 pages) Paper No: VIB-17-1336; doi: 10.1115/1.4039536 History: Received July 24, 2017; Revised February 22, 2018

In Part 2 of the two-paper series, the asymmetrically laminated piezoelectric shell subjected to distributed bias voltage as modeled in Part 1 is analytically and numerically investigated. Three out-of-plane degrees-of-freedom (DOFs) and a number of in-plane DOFs are retained to study the shell's snap-through phenomenon. A convergence study first confirms that the number of the in-plane DOFs retained affects not only the number of predicted equilibrium states when the bias voltage is absent but also the prediction of the critical bias voltage for snap-through to occur and the types of snap-through mechanisms. Equilibrium states can be symmetric or asymmetric, involving only a symmetric out-of-plane DOF, and additional asymmetric out-of-plane DOFs, respectively. For symmetric equilibrium states, the snap-through mechanism can evolve from the classical bidirectional snap-through and latching to a new type of snap-through that only allows snap-through in one direction (i.e., unidirectional snap-through), depending on the distribution of the bias voltage. For asymmetric equilibrium states, degeneration can occur to the asymmetric bifurcation points when the radii of curvature are equal. Finally, the unidirectional snap-through renders an explanation to the experimental findings in Part 1.

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References

Tai, W. C. , Luo, C. , Yang, C. W. , Cao, G. , and Shen, I. S. , 2018, “ Voltage-Induced Snap-Through of an Asymmetrically Laminated, Piezoelectric, Thin-Film Diaphragm Micro-Actuator: Part 1—Experimental Studies and Mathematical Modeling,” ASME J. Vib. Acoust., accepted.
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Luo, C. , Tai, W. , Yang, C.-W. , Cao, G. , and Shen, I. , 2016, “ Effects of Added Mass on Lead-Zirconate-Titanate Thin-Film Microactuators in Aqueous Environments,” ASME J. Vib. Acoust., 138(6), p. 061015. [CrossRef]
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Figures

Grahic Jump Location
Fig. 1

Convergence study. (a) Contour plots of Δ(R̂x,R̂y)=0 of three cases of in-plane shape function number indicator N: (I) N = 2, (II) N = 3, and (III) N = 4. In each case, admissible radii of curvature are enclosed by the contour. (b) Relative error ϵN of the snap-through voltage with respect to N.

Grahic Jump Location
Fig. 2

Voltage-response diagrams with R̂x=R̂y=16 and Π=−0.48: (a) N = 2, (b) N = 3, (c) N = 4, and (d) N = 6. A and C are stable equilibrium states, whereas B is the unstable equilibrium state. S is the limit point.

Grahic Jump Location
Fig. 3

Voltage-response diagrams with (a) Π=−0.45, (b) Π=−0.48, (c) Π=−0.58, and (d) the deformed shape along the equilibrium curve in (b). Note that (b) resembles Fig. 7 in Ref. [7]. Also note that x and r1 in (d) are in physical unit of μm.

Grahic Jump Location
Fig. 4

Voltage-response diagrams of symmetric snap-through and asymmetric snap-through of first kind and second kind. (a) Equal radii of curvature Rx=Ry=16. (b) Unequal radii of curvature Rx = 16 and Ry=16.1. S: limit point; AS: bifurcation point. The superscripts a, b, and c correspond to the branches of type (a), (b), and (c), respectively.

Grahic Jump Location
Fig. 5

Equilibrium diagrams of asymmetric snap-through of first kind (first row) and of second kind (second row). (a) Equal radii of curvature R̂x=R̂y=16 of type (b). (b) Unequal radii of curvature Rx = 16 and Ry=16.1 of type (b). (c) Equal radii of curvature R̂x=R̂y=16 of type (c). (d) Unequal radii of curvature Rx = 16 and Ry=16.1 of type (c). S: limit point; AS: bifurcation point. The superscripts a, b, and c correspond to the branches of type (a), (b), and (c), respectively.

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