Bistable Cantilevers Actuated by Fringing Electrostatic Fields

[+] Author and Article Information
Naftaly Krakover

Faculty of Engineering,
School of Mechanical Engineering,
Tel Aviv University,
Ramat Aviv,
Tel Aviv 69978, Israel
e-mail: Naftalyk@mail.tau.ac.il

Slava Krylov

Faculty of Engineering,
School of Mechanical Engineering,
Tel Aviv University,
Ramat Aviv,
Tel Aviv 69978, Israel
e-mail: krylov@tauex.tau.ac.il

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received December 16, 2016; final manuscript received April 19, 2017; published online May 30, 2017. Assoc. Editor: John Judge.

J. Vib. Acoust 139(4), 040908 (May 30, 2017) (10 pages) Paper No: VIB-16-1598; doi: 10.1115/1.4036625 History: Received December 16, 2016; Revised April 19, 2017

Bistable microstructures are distinguished by their ability to stay in two different stable configurations at the same loading. They manifest rich behavior and are advantageous in applications such as switches, nonvolatile memories, and sensors. Bistability of initially curved or buckled double-clamped beams, curved plates, and shells is associated with mechanical geometric nonlinearity appearing due to coupling between bending and compressive axial/in-plane stress. The bistable behavior is achieved by using a combination of carefully tailored initial shape and constrained boundaries. However, these statically indeterminate structures suffer from high sensitivity to temperature and residual stress. In this work, we show using the model that by combining electrostatic actuation by fringing fields with direct transversal forcing by a parallel-plate electrode or piezoelectric (PZT) transducer, bistable behavior can be obtained in a simple cantilever structure distinguished by robustness and low thermal sensitivity. Reduced-order model of the cantilever was built using Galerkin decomposition, the electrostatic force was obtained by means of three-dimensional (3D) finite elements (FEs) modeling. We also demonstrate that operation of the device in the vicinity of the bistability threshold may enhance the frequency sensitivity of the cantilever to loading. This sensitivity-enhancement approach may have applications in a broad range of resonant microelectromechanical inertial, force, mass, and biosensors as well as in atomic force microscopy (AFM).

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Grahic Jump Location
Fig. 1

Schematics of a cantilever actuated by fringing electrostatic fields: (a) initial undeformed configuration and (b) deformed configuration. Insets illustrate the schematic view of the electrostatic fringing fields for each case.

Grahic Jump Location
Fig. 2

Schematics of the mechanical restoring force, electrostatic fringing fields' restoring force, and the combination of both, as a function of the deflection of the beam

Grahic Jump Location
Fig. 3

Finite elements results—the electric potential for differing vertical displacements of the movable electrode (the beam) (a) undeformed configuration and (b) deformed configuration corresponding to the vertical displacement of the beam of 10 μm. Horizontal y and vertical z coordinates are shown in μm. The voltage of 1 V is applied to the side electrode.

Grahic Jump Location
Fig. 4

Force per unit length of the beam—FE results (markers) and the fit Eq. (2) (solid line). The parameters of the beam are shown in Table 1, the voltage applied to the electrode is 1 V.

Grahic Jump Location
Fig. 5

Equilibrium curves for the different values of the direct current voltage VS (numbers) applied to the side electrode. The curve corresponding to Vs = 90 V reflects bistable behavior. Solid lines correspond to the stable solution, dashed line depicts unstable branch of the equilibrium path.

Grahic Jump Location
Fig. 6

Equilibrium curves for the beam obtained using different models: single-DOF RO model (thin solid line), collocation method (dashed line), 3D FE model (markers)

Grahic Jump Location
Fig. 7

Cantilever actuated by a parallel-plate electrode in addition to the fringing field side electrode

Grahic Jump Location
Fig. 8

Model results—equilibrium curves for the beam simultaneously actuated by a parallel-plate electrode of the length Lpp = L/2 and by the side fringing fields' electrode. Different curves correspond to different voltages VS applied on the sidefringing field electrode. Solid line corresponds to stable equilibrium configurations, dashed line represents unstable equilibrium.

Grahic Jump Location
Fig. 9

Cantilever made from two layers—a passive silicon layer and an active PZT layer. Electrostatic fringing field is provided by the side electrode. Inset shows the cross section of the beam.

Grahic Jump Location
Fig. 10

Equilibrium curves—model results, piezoelectric actuation. Different curves correspond to different fringing fields' electrostatic force parameterized by VS applied to the side electrode.

Grahic Jump Location
Fig. 11

Fundamental mode frequency of the cantilever as a function of the transverse “mechanical” loading. Different curves correspond to the different values of the voltage VS (numbers) applied to the side fringing field electrode. Discontinuity can be seen on the curve corresponding to VS = 90 V, which corresponds to the bistable behavior (see Fig. 5).



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